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(163,567)] 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("567.163"); 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([0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 567 = 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 567.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == 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: 8.0.1767277521.3
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + x^{6} - 10x^{5} + 38x^{4} - 40x^{3} + 64x^{2} - 38x + 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 487.1
Root \(-1.54162 + 1.88572i\) of defining polynomial
Character \(\chi\) \(=\) 567.487
Dual form 567.2.e.c.163.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.10400 - 1.91218i) q^{2} +(-1.43762 + 2.49004i) q^{4} +(1.90389 + 3.29764i) q^{5} +(1.82854 - 1.91218i) q^{7} +1.93254 q^{8} +(4.20379 - 7.28117i) q^{10} +(-2.16217 + 3.74498i) q^{11} -2.87525 q^{13} +(-5.67514 - 1.38546i) q^{14} +(0.741726 + 1.28471i) q^{16} +(-2.01297 + 3.48657i) q^{17} +(0.804103 + 1.39275i) q^{19} -10.9483 q^{20} +9.54811 q^{22} +(1.33363 + 2.30991i) q^{23} +(-4.74962 + 8.22658i) q^{25} +(3.17427 + 5.49799i) q^{26} +(2.13264 + 7.30213i) q^{28} +0.750492 q^{29} +(-0.0702679 + 0.121708i) q^{31} +(3.57027 - 6.18389i) q^{32} +8.88928 q^{34} +(9.78703 + 2.38928i) q^{35} +(4.14141 + 7.17313i) q^{37} +(1.77546 - 3.07518i) q^{38} +(3.67935 + 6.37282i) q^{40} -10.3745 q^{41} +0.267040 q^{43} +(-6.21676 - 10.7677i) q^{44} +(2.94464 - 5.10026i) q^{46} +(3.96627 + 6.86978i) q^{47} +(-0.312869 - 6.99300i) q^{49} +20.9743 q^{50} +(4.13352 - 7.15947i) q^{52} +(5.61189 - 9.72008i) q^{53} -16.4661 q^{55} +(3.53373 - 3.69537i) q^{56} +(-0.828542 - 1.43508i) q^{58} +(0.346599 - 0.600327i) q^{59} +(-1.05372 - 1.82510i) q^{61} +0.310302 q^{62} -12.7994 q^{64} +(-5.47416 - 9.48152i) q^{65} +(5.38314 - 9.32387i) q^{67} +(-5.78780 - 10.0248i) q^{68} +(-6.23612 - 21.3523i) q^{70} +3.62399 q^{71} +(1.78756 - 3.09614i) q^{73} +(9.14422 - 15.8383i) q^{74} -4.62399 q^{76} +(3.20747 + 10.9823i) q^{77} +(7.71168 + 13.3570i) q^{79} +(-2.82433 + 4.89189i) q^{80} +(11.4534 + 19.8379i) q^{82} +6.44067 q^{83} -15.3299 q^{85} +(-0.294812 - 0.510629i) q^{86} +(-4.17847 + 7.23733i) q^{88} +(-0.128437 - 0.222459i) q^{89} +(-5.25751 + 5.49799i) q^{91} -7.66900 q^{92} +(8.75751 - 15.1684i) q^{94} +(-3.06185 + 5.30328i) q^{95} +1.05856 q^{97} +(-13.0265 + 8.31853i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - q^{2} - 5 q^{4} - 2 q^{5} + q^{7} - 6 q^{8} + 7 q^{10} - 5 q^{11} - 10 q^{13} - 23 q^{14} + q^{16} - 6 q^{17} + 8 q^{19} - 16 q^{20} - 14 q^{22} + 12 q^{23} - 8 q^{25} - q^{26} + 5 q^{28} - 20 q^{29}+ \cdots + 11 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.10400 − 1.91218i −0.780644 − 1.35212i −0.931567 − 0.363570i \(-0.881558\pi\)
0.150922 − 0.988546i \(-0.451776\pi\)
\(3\) 0 0
\(4\) −1.43762 + 2.49004i −0.718812 + 1.24502i
\(5\) 1.90389 + 3.29764i 0.851447 + 1.47475i 0.879903 + 0.475154i \(0.157608\pi\)
−0.0284558 + 0.999595i \(0.509059\pi\)
\(6\) 0 0
\(7\) 1.82854 − 1.91218i 0.691124 − 0.722736i
\(8\) 1.93254 0.683256
\(9\) 0 0
\(10\) 4.20379 − 7.28117i 1.32935 − 2.30251i
\(11\) −2.16217 + 3.74498i −0.651918 + 1.12915i 0.330739 + 0.943722i \(0.392702\pi\)
−0.982657 + 0.185433i \(0.940631\pi\)
\(12\) 0 0
\(13\) −2.87525 −0.797450 −0.398725 − 0.917071i \(-0.630547\pi\)
−0.398725 + 0.917071i \(0.630547\pi\)
\(14\) −5.67514 − 1.38546i −1.51675 − 0.370279i
\(15\) 0 0
\(16\) 0.741726 + 1.28471i 0.185432 + 0.321177i
\(17\) −2.01297 + 3.48657i −0.488218 + 0.845618i −0.999908 − 0.0135517i \(-0.995686\pi\)
0.511690 + 0.859170i \(0.329020\pi\)
\(18\) 0 0
\(19\) 0.804103 + 1.39275i 0.184474 + 0.319518i 0.943399 − 0.331660i \(-0.107609\pi\)
−0.758925 + 0.651178i \(0.774275\pi\)
\(20\) −10.9483 −2.44812
\(21\) 0 0
\(22\) 9.54811 2.03566
\(23\) 1.33363 + 2.30991i 0.278080 + 0.481649i 0.970908 − 0.239455i \(-0.0769686\pi\)
−0.692828 + 0.721103i \(0.743635\pi\)
\(24\) 0 0
\(25\) −4.74962 + 8.22658i −0.949923 + 1.64532i
\(26\) 3.17427 + 5.49799i 0.622525 + 1.07824i
\(27\) 0 0
\(28\) 2.13264 + 7.30213i 0.403032 + 1.37997i
\(29\) 0.750492 0.139363 0.0696815 − 0.997569i \(-0.477802\pi\)
0.0696815 + 0.997569i \(0.477802\pi\)
\(30\) 0 0
\(31\) −0.0702679 + 0.121708i −0.0126205 + 0.0218593i −0.872267 − 0.489031i \(-0.837351\pi\)
0.859646 + 0.510890i \(0.170684\pi\)
\(32\) 3.57027 − 6.18389i 0.631140 − 1.09317i
\(33\) 0 0
\(34\) 8.88928 1.52450
\(35\) 9.78703 + 2.38928i 1.65431 + 0.403863i
\(36\) 0 0
\(37\) 4.14141 + 7.17313i 0.680844 + 1.17926i 0.974724 + 0.223414i \(0.0717201\pi\)
−0.293880 + 0.955842i \(0.594947\pi\)
\(38\) 1.77546 − 3.07518i 0.288017 − 0.498860i
\(39\) 0 0
\(40\) 3.67935 + 6.37282i 0.581756 + 1.00763i
\(41\) −10.3745 −1.62022 −0.810111 − 0.586277i \(-0.800593\pi\)
−0.810111 + 0.586277i \(0.800593\pi\)
\(42\) 0 0
\(43\) 0.267040 0.0407232 0.0203616 − 0.999793i \(-0.493518\pi\)
0.0203616 + 0.999793i \(0.493518\pi\)
\(44\) −6.21676 − 10.7677i −0.937212 − 1.62330i
\(45\) 0 0
\(46\) 2.94464 − 5.10026i 0.434163 − 0.751993i
\(47\) 3.96627 + 6.86978i 0.578540 + 1.00206i 0.995647 + 0.0932032i \(0.0297106\pi\)
−0.417107 + 0.908857i \(0.636956\pi\)
\(48\) 0 0
\(49\) −0.312869 − 6.99300i −0.0446956 − 0.999001i
\(50\) 20.9743 2.96621
\(51\) 0 0
\(52\) 4.13352 − 7.15947i 0.573216 − 0.992839i
\(53\) 5.61189 − 9.72008i 0.770852 − 1.33516i −0.166244 − 0.986085i \(-0.553164\pi\)
0.937096 − 0.349071i \(-0.113503\pi\)
\(54\) 0 0
\(55\) −16.4661 −2.22029
\(56\) 3.53373 − 3.69537i 0.472215 − 0.493814i
\(57\) 0 0
\(58\) −0.828542 − 1.43508i −0.108793 − 0.188435i
\(59\) 0.346599 − 0.600327i 0.0451234 − 0.0781560i −0.842582 − 0.538569i \(-0.818965\pi\)
0.887705 + 0.460413i \(0.152299\pi\)
\(60\) 0 0
\(61\) −1.05372 − 1.82510i −0.134915 − 0.233680i 0.790650 − 0.612268i \(-0.209743\pi\)
−0.925565 + 0.378589i \(0.876409\pi\)
\(62\) 0.310302 0.0394085
\(63\) 0 0
\(64\) −12.7994 −1.59992
\(65\) −5.47416 − 9.48152i −0.678986 − 1.17604i
\(66\) 0 0
\(67\) 5.38314 − 9.32387i 0.657655 − 1.13909i −0.323566 − 0.946206i \(-0.604882\pi\)
0.981221 − 0.192886i \(-0.0617848\pi\)
\(68\) −5.78780 − 10.0248i −0.701873 − 1.21568i
\(69\) 0 0
\(70\) −6.23612 − 21.3523i −0.745359 − 2.55209i
\(71\) 3.62399 0.430088 0.215044 − 0.976604i \(-0.431010\pi\)
0.215044 + 0.976604i \(0.431010\pi\)
\(72\) 0 0
\(73\) 1.78756 − 3.09614i 0.209217 − 0.362375i −0.742251 − 0.670122i \(-0.766242\pi\)
0.951468 + 0.307747i \(0.0995750\pi\)
\(74\) 9.14422 − 15.8383i 1.06299 − 1.84116i
\(75\) 0 0
\(76\) −4.62399 −0.530408
\(77\) 3.20747 + 10.9823i 0.365525 + 1.25155i
\(78\) 0 0
\(79\) 7.71168 + 13.3570i 0.867632 + 1.50278i 0.864410 + 0.502787i \(0.167692\pi\)
0.00322152 + 0.999995i \(0.498975\pi\)
\(80\) −2.82433 + 4.89189i −0.315770 + 0.546930i
\(81\) 0 0
\(82\) 11.4534 + 19.8379i 1.26482 + 2.19073i
\(83\) 6.44067 0.706956 0.353478 − 0.935443i \(-0.384999\pi\)
0.353478 + 0.935443i \(0.384999\pi\)
\(84\) 0 0
\(85\) −15.3299 −1.66277
\(86\) −0.294812 − 0.510629i −0.0317904 − 0.0550625i
\(87\) 0 0
\(88\) −4.17847 + 7.23733i −0.445427 + 0.771502i
\(89\) −0.128437 − 0.222459i −0.0136143 − 0.0235806i 0.859138 − 0.511744i \(-0.171000\pi\)
−0.872752 + 0.488163i \(0.837667\pi\)
\(90\) 0 0
\(91\) −5.25751 + 5.49799i −0.551137 + 0.576346i
\(92\) −7.66900 −0.799549
\(93\) 0 0
\(94\) 8.75751 − 15.1684i 0.903268 − 1.56451i
\(95\) −3.06185 + 5.30328i −0.314139 + 0.544105i
\(96\) 0 0
\(97\) 1.05856 0.107481 0.0537403 − 0.998555i \(-0.482886\pi\)
0.0537403 + 0.998555i \(0.482886\pi\)
\(98\) −13.0265 + 8.31853i −1.31587 + 0.840298i
\(99\) 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.e.c.487.1 yes 8
3.2 odd 2 567.2.e.d.487.4 yes 8
7.2 even 3 inner 567.2.e.c.163.1 ✓ 8
7.3 odd 6 3969.2.a.w.1.4 4
7.4 even 3 3969.2.a.x.1.4 4
9.2 odd 6 567.2.g.k.109.4 8
9.4 even 3 567.2.h.k.298.4 8
9.5 odd 6 567.2.h.j.298.1 8
9.7 even 3 567.2.g.j.109.1 8
21.2 odd 6 567.2.e.d.163.4 yes 8
21.11 odd 6 3969.2.a.s.1.1 4
21.17 even 6 3969.2.a.t.1.1 4
63.2 odd 6 567.2.h.j.352.1 8
63.16 even 3 567.2.h.k.352.4 8
63.23 odd 6 567.2.g.k.541.4 8
63.58 even 3 567.2.g.j.541.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
567.2.e.c.163.1 ✓ 8 7.2 even 3 inner
567.2.e.c.487.1 yes 8 1.1 even 1 trivial
567.2.e.d.163.4 yes 8 21.2 odd 6
567.2.e.d.487.4 yes 8 3.2 odd 2
567.2.g.j.109.1 8 9.7 even 3
567.2.g.j.541.1 8 63.58 even 3
567.2.g.k.109.4 8 9.2 odd 6
567.2.g.k.541.4 8 63.23 odd 6
567.2.h.j.298.1 8 9.5 odd 6
567.2.h.j.352.1 8 63.2 odd 6
567.2.h.k.298.4 8 9.4 even 3
567.2.h.k.352.4 8 63.16 even 3
3969.2.a.s.1.1 4 21.11 odd 6
3969.2.a.t.1.1 4 21.17 even 6
3969.2.a.w.1.4 4 7.3 odd 6
3969.2.a.x.1.4 4 7.4 even 3