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(298,567)] 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("567.298"); 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([4, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 567 = 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 567.h (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,-2,0,10,2,0,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == 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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 298.1
Root \(-1.54162 - 1.88572i\) of defining polynomial
Character \(\chi\) \(=\) 567.298
Dual form 567.2.h.j.352.1

$q$-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)
 
\(f(q)\) \(=\) \(q-2.20800 q^{2} +2.87525 q^{4} +(-1.90389 + 3.29764i) q^{5} +(0.741726 + 2.53965i) q^{7} -1.93254 q^{8} +(4.20379 - 7.28117i) q^{10} +(2.16217 + 3.74498i) q^{11} +(1.43762 + 2.49004i) q^{13} +(-1.63773 - 5.60755i) q^{14} -1.48345 q^{16} +(2.01297 - 3.48657i) q^{17} +(0.804103 + 1.39275i) q^{19} +(-5.47416 + 9.48152i) q^{20} +(-4.77406 - 8.26891i) q^{22} +(-1.33363 + 2.30991i) q^{23} +(-4.74962 - 8.22658i) q^{25} +(-3.17427 - 5.49799i) q^{26} +(2.13264 + 7.30213i) q^{28} +(0.375246 - 0.649945i) q^{29} +0.140536 q^{31} +7.14054 q^{32} +(-4.44464 + 7.69834i) q^{34} +(-9.78703 - 2.38928i) q^{35} +(4.14141 + 7.17313i) q^{37} +(-1.77546 - 3.07518i) q^{38} +(3.67935 - 6.37282i) q^{40} +(-5.18724 - 8.98456i) q^{41} +(-0.133520 + 0.231264i) q^{43} +(6.21676 + 10.7677i) q^{44} +(2.94464 - 5.10026i) q^{46} +7.93254 q^{47} +(-5.89969 + 3.76745i) q^{49} +(10.4871 + 18.1643i) q^{50} +(4.13352 + 7.15947i) q^{52} +(-5.61189 + 9.72008i) q^{53} -16.4661 q^{55} +(-1.43342 - 4.90798i) q^{56} +(-0.828542 + 1.43508i) q^{58} +0.693198 q^{59} +2.10744 q^{61} -0.310302 q^{62} -12.7994 q^{64} -10.9483 q^{65} -10.7663 q^{67} +(5.78780 - 10.0248i) q^{68} +(21.6097 + 5.27553i) q^{70} -3.62399 q^{71} +(1.78756 - 3.09614i) q^{73} +(-9.14422 - 15.8383i) q^{74} +(2.31199 + 4.00449i) q^{76} +(-7.90723 + 8.26891i) q^{77} -15.4234 q^{79} +(2.82433 - 4.89189i) q^{80} +(11.4534 + 19.8379i) q^{82} +(3.22034 - 5.57779i) q^{83} +(7.66497 + 13.2761i) q^{85} +(0.294812 - 0.510629i) q^{86} +(-4.17847 - 7.23733i) q^{88} +(0.128437 + 0.222459i) q^{89} +(-5.25751 + 5.49799i) q^{91} +(-3.83450 + 6.64155i) q^{92} -17.5150 q^{94} -6.12370 q^{95} +(-0.529281 + 0.916742i) q^{97} +(13.0265 - 8.31853i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{2} + 10 q^{4} + 2 q^{5} + q^{7} + 6 q^{8} + 7 q^{10} + 5 q^{11} + 5 q^{13} - 16 q^{14} - 2 q^{16} + 6 q^{17} + 8 q^{19} - 8 q^{20} + 7 q^{22} - 12 q^{23} - 8 q^{25} + q^{26} + 5 q^{28} - 10 q^{29}+ \cdots - 11 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.20800 −1.56129 −0.780644 − 0.624975i \(-0.785109\pi\)
−0.780644 + 0.624975i \(0.785109\pi\)
\(3\) 0 0
\(4\) 2.87525 1.43762
\(5\) −1.90389 + 3.29764i −0.851447 + 1.47475i 0.0284558 + 0.999595i \(0.490941\pi\)
−0.879903 + 0.475154i \(0.842392\pi\)
\(6\) 0 0
\(7\) 0.741726 + 2.53965i 0.280346 + 0.959899i
\(8\) −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 0.982657 + 0.185433i \(0.0593687\pi\)
−0.330739 + 0.943722i \(0.607298\pi\)
\(12\) 0 0
\(13\) 1.43762 + 2.49004i 0.398725 + 0.690612i 0.993569 − 0.113229i \(-0.0361195\pi\)
−0.594844 + 0.803841i \(0.702786\pi\)
\(14\) −1.63773 − 5.60755i −0.437701 − 1.49868i
\(15\) 0 0
\(16\) −1.48345 −0.370863
\(17\) 2.01297 − 3.48657i 0.488218 − 0.845618i −0.511690 − 0.859170i \(-0.670980\pi\)
0.999908 + 0.0135517i \(0.00431378\pi\)
\(18\) 0 0
\(19\) 0.804103 + 1.39275i 0.184474 + 0.319518i 0.943399 − 0.331660i \(-0.107609\pi\)
−0.758925 + 0.651178i \(0.774275\pi\)
\(20\) −5.47416 + 9.48152i −1.22406 + 2.12013i
\(21\) 0 0
\(22\) −4.77406 − 8.26891i −1.01783 − 1.76294i
\(23\) −1.33363 + 2.30991i −0.278080 + 0.481649i −0.970908 − 0.239455i \(-0.923031\pi\)
0.692828 + 0.721103i \(0.256365\pi\)
\(24\) 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
\(29\) 0.375246 − 0.649945i 0.0696815 − 0.120692i −0.829080 − 0.559131i \(-0.811135\pi\)
0.898761 + 0.438439i \(0.144468\pi\)
\(30\) 0 0
\(31\) 0.140536 0.0252410 0.0126205 − 0.999920i \(-0.495983\pi\)
0.0126205 + 0.999920i \(0.495983\pi\)
\(32\) 7.14054 1.26228
\(33\) 0 0
\(34\) −4.44464 + 7.69834i −0.762249 + 1.32025i
\(35\) −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
\(41\) −5.18724 − 8.98456i −0.810111 − 1.40315i −0.912786 − 0.408438i \(-0.866074\pi\)
0.102675 − 0.994715i \(-0.467260\pi\)
\(42\) 0 0
\(43\) −0.133520 + 0.231264i −0.0203616 + 0.0352674i −0.876027 − 0.482263i \(-0.839815\pi\)
0.855665 + 0.517530i \(0.173148\pi\)
\(44\) 6.21676 + 10.7677i 0.937212 + 1.62330i
\(45\) 0 0
\(46\) 2.94464 − 5.10026i 0.434163 − 0.751993i
\(47\) 7.93254 1.15708 0.578540 − 0.815654i \(-0.303623\pi\)
0.578540 + 0.815654i \(0.303623\pi\)
\(48\) 0 0
\(49\) −5.89969 + 3.76745i −0.842812 + 0.538208i
\(50\) 10.4871 + 18.1643i 1.48310 + 2.56881i
\(51\) 0 0
\(52\) 4.13352 + 7.15947i 0.573216 + 0.992839i
\(53\) −5.61189 + 9.72008i −0.770852 + 1.33516i 0.166244 + 0.986085i \(0.446836\pi\)
−0.937096 + 0.349071i \(0.886497\pi\)
\(54\) 0 0
\(55\) −16.4661 −2.22029
\(56\) −1.43342 − 4.90798i −0.191548 − 0.655857i
\(57\) 0 0
\(58\) −0.828542 + 1.43508i −0.108793 + 0.188435i
\(59\) 0.693198 0.0902468 0.0451234 − 0.998981i \(-0.485632\pi\)
0.0451234 + 0.998981i \(0.485632\pi\)
\(60\) 0 0
\(61\) 2.10744 0.269830 0.134915 − 0.990857i \(-0.456924\pi\)
0.134915 + 0.990857i \(0.456924\pi\)
\(62\) −0.310302 −0.0394085
\(63\) 0 0
\(64\) −12.7994 −1.59992
\(65\) −10.9483 −1.35797
\(66\) 0 0
\(67\) −10.7663 −1.31531 −0.657655 − 0.753319i \(-0.728451\pi\)
−0.657655 + 0.753319i \(0.728451\pi\)
\(68\) 5.78780 − 10.0248i 0.701873 − 1.21568i
\(69\) 0 0
\(70\) 21.6097 + 5.27553i 2.58286 + 0.630547i
\(71\) −3.62399 −0.430088 −0.215044 − 0.976604i \(-0.568990\pi\)
−0.215044 + 0.976604i \(0.568990\pi\)
\(72\) 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
\(76\) 2.31199 + 4.00449i 0.265204 + 0.459347i
\(77\) −7.90723 + 8.26891i −0.901112 + 0.942329i
\(78\) 0 0
\(79\) −15.4234 −1.73526 −0.867632 − 0.497207i \(-0.834359\pi\)
−0.867632 + 0.497207i \(0.834359\pi\)
\(80\) 2.82433 − 4.89189i 0.315770 − 0.546930i
\(81\) 0 0
\(82\) 11.4534 + 19.8379i 1.26482 + 2.19073i
\(83\) 3.22034 − 5.57779i 0.353478 − 0.612241i −0.633378 − 0.773842i \(-0.718332\pi\)
0.986856 + 0.161601i \(0.0516657\pi\)
\(84\) 0 0
\(85\) 7.66497 + 13.2761i 0.831383 + 1.44000i
\(86\) 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 0.872752 − 0.488163i \(-0.162333\pi\)
−0.859138 + 0.511744i \(0.829000\pi\)
\(90\) 0 0
\(91\) −5.25751 + 5.49799i −0.551137 + 0.576346i
\(92\) −3.83450 + 6.64155i −0.399774 + 0.692429i
\(93\) 0 0
\(94\) −17.5150 −1.80654
\(95\) −6.12370 −0.628279
\(96\) 0 0
\(97\) −0.529281 + 0.916742i −0.0537403 + 0.0930810i −0.891644 − 0.452737i \(-0.850448\pi\)
0.837904 + 0.545818i \(0.183781\pi\)
\(98\) 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.h.j.298.1 8
3.2 odd 2 567.2.h.k.298.4 8
7.2 even 3 567.2.g.k.541.4 8
9.2 odd 6 567.2.e.c.487.1 yes 8
9.4 even 3 567.2.g.k.109.4 8
9.5 odd 6 567.2.g.j.109.1 8
9.7 even 3 567.2.e.d.487.4 yes 8
21.2 odd 6 567.2.g.j.541.1 8
63.2 odd 6 567.2.e.c.163.1 ✓ 8
63.11 odd 6 3969.2.a.x.1.4 4
63.16 even 3 567.2.e.d.163.4 yes 8
63.23 odd 6 567.2.h.k.352.4 8
63.25 even 3 3969.2.a.s.1.1 4
63.38 even 6 3969.2.a.w.1.4 4
63.52 odd 6 3969.2.a.t.1.1 4
63.58 even 3 inner 567.2.h.j.352.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
567.2.e.c.163.1 ✓ 8 63.2 odd 6
567.2.e.c.487.1 yes 8 9.2 odd 6
567.2.e.d.163.4 yes 8 63.16 even 3
567.2.e.d.487.4 yes 8 9.7 even 3
567.2.g.j.109.1 8 9.5 odd 6
567.2.g.j.541.1 8 21.2 odd 6
567.2.g.k.109.4 8 9.4 even 3
567.2.g.k.541.4 8 7.2 even 3
567.2.h.j.298.1 8 1.1 even 1 trivial
567.2.h.j.352.1 8 63.58 even 3 inner
567.2.h.k.298.4 8 3.2 odd 2
567.2.h.k.352.4 8 63.23 odd 6
3969.2.a.s.1.1 4 63.25 even 3
3969.2.a.t.1.1 4 63.52 odd 6
3969.2.a.w.1.4 4 63.38 even 6
3969.2.a.x.1.4 4 63.11 odd 6