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(80,567)] 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("567.80"); 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([3, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 567 = 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 567.p (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [32,0,0,16,0,0,-8,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.52751779461\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 80.8
Character \(\chi\) \(=\) 567.80
Dual form 567.2.p.e.404.8

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.118865 + 0.0686265i) q^{2} +(-0.990581 + 1.71574i) q^{4} +(-1.86818 - 3.23578i) q^{5} +(1.08649 + 2.41237i) q^{7} -0.546426i q^{8} +(0.444120 + 0.256413i) q^{10} +(2.32679 + 1.34337i) q^{11} +0.100488i q^{13} +(-0.294697 - 0.212184i) q^{14} +(-1.94366 - 3.36652i) q^{16} +(-1.56658 + 2.71340i) q^{17} +(-5.70035 + 3.29110i) q^{19} +7.40232 q^{20} -0.368763 q^{22} +(-4.24223 + 2.44925i) q^{23} +(-4.48018 + 7.75989i) q^{25} +(-0.00689613 - 0.0119444i) q^{26} +(-5.21525 - 0.525527i) q^{28} +8.79424i q^{29} +(1.44974 + 0.837009i) q^{31} +(1.40850 + 0.813199i) q^{32} -0.430037i q^{34} +(5.77616 - 8.02237i) q^{35} +(4.41395 + 7.64519i) q^{37} +(0.451713 - 0.782390i) q^{38} +(-1.76811 + 1.02082i) q^{40} -6.74011 q^{41} +0.631237 q^{43} +(-4.60974 + 2.66144i) q^{44} +(0.336167 - 0.582258i) q^{46} +(-3.14006 - 5.43875i) q^{47} +(-4.63910 + 5.24202i) q^{49} -1.22983i q^{50} +(-0.172411 - 0.0995413i) q^{52} +(-1.05449 - 0.608812i) q^{53} -10.0386i q^{55} +(1.31818 - 0.593684i) q^{56} +(-0.603518 - 1.04532i) q^{58} +(5.43266 - 9.40964i) q^{59} +(5.52604 - 3.19046i) q^{61} -0.229764 q^{62} +7.55142 q^{64} +(0.325156 - 0.187729i) q^{65} +(-3.15343 + 5.46190i) q^{67} +(-3.10366 - 5.37569i) q^{68} +(-0.136033 + 1.34997i) q^{70} -2.25070i q^{71} +(5.76973 + 3.33116i) q^{73} +(-1.04932 - 0.605828i) q^{74} -13.0404i q^{76} +(-0.712691 + 7.07264i) q^{77} +(-1.79041 - 3.10109i) q^{79} +(-7.26221 + 12.5785i) q^{80} +(0.801160 - 0.462550i) q^{82} -7.50252 q^{83} +11.7066 q^{85} +(-0.0750317 + 0.0433195i) q^{86} +(0.734053 - 1.27142i) q^{88} +(4.83431 + 8.37327i) q^{89} +(-0.242414 + 0.109179i) q^{91} -9.70473i q^{92} +(0.746485 + 0.430983i) q^{94} +(21.2985 + 12.2967i) q^{95} -9.61294i q^{97} +(0.191682 - 0.941455i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q + 16 q^{4} - 8 q^{7} - 28 q^{16} + 24 q^{22} - 16 q^{25} - 16 q^{28} - 48 q^{31} - 4 q^{37} + 56 q^{43} + 12 q^{46} - 4 q^{49} + 48 q^{52} + 36 q^{58} + 12 q^{61} - 80 q^{64} - 20 q^{67} + 120 q^{70}+ \cdots + 72 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(325\) \(407\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.118865 + 0.0686265i −0.0840499 + 0.0485262i −0.541436 − 0.840742i \(-0.682119\pi\)
0.457386 + 0.889268i \(0.348786\pi\)
\(3\) 0 0
\(4\) −0.990581 + 1.71574i −0.495290 + 0.857868i
\(5\) −1.86818 − 3.23578i −0.835475 − 1.44708i −0.893643 − 0.448778i \(-0.851860\pi\)
0.0581689 − 0.998307i \(-0.481474\pi\)
\(6\) 0 0
\(7\) 1.08649 + 2.41237i 0.410653 + 0.911792i
\(8\) − 0.546426i − 0.193191i
\(9\) 0 0
\(10\) 0.444120 + 0.256413i 0.140443 + 0.0810849i
\(11\) 2.32679 + 1.34337i 0.701553 + 0.405042i 0.807925 − 0.589285i \(-0.200590\pi\)
−0.106373 + 0.994326i \(0.533924\pi\)
\(12\) 0 0
\(13\) 0.100488i 0.0278703i 0.999903 + 0.0139352i \(0.00443584\pi\)
−0.999903 + 0.0139352i \(0.995564\pi\)
\(14\) −0.294697 − 0.212184i −0.0787612 − 0.0567085i
\(15\) 0 0
\(16\) −1.94366 − 3.36652i −0.485916 − 0.841630i
\(17\) −1.56658 + 2.71340i −0.379953 + 0.658097i −0.991055 − 0.133454i \(-0.957393\pi\)
0.611102 + 0.791552i \(0.290726\pi\)
\(18\) 0 0
\(19\) −5.70035 + 3.29110i −1.30775 + 0.755030i −0.981720 − 0.190329i \(-0.939045\pi\)
−0.326030 + 0.945359i \(0.605711\pi\)
\(20\) 7.40232 1.65521
\(21\) 0 0
\(22\) −0.368763 −0.0786206
\(23\) −4.24223 + 2.44925i −0.884566 + 0.510704i −0.872161 − 0.489219i \(-0.837282\pi\)
−0.0124046 + 0.999923i \(0.503949\pi\)
\(24\) 0 0
\(25\) −4.48018 + 7.75989i −0.896035 + 1.55198i
\(26\) −0.00689613 − 0.0119444i −0.00135244 − 0.00234250i
\(27\) 0 0
\(28\) −5.21525 − 0.525527i −0.985590 − 0.0993153i
\(29\) 8.79424i 1.63305i 0.577310 + 0.816525i \(0.304102\pi\)
−0.577310 + 0.816525i \(0.695898\pi\)
\(30\) 0 0
\(31\) 1.44974 + 0.837009i 0.260381 + 0.150331i 0.624509 − 0.781018i \(-0.285299\pi\)
−0.364127 + 0.931349i \(0.618633\pi\)
\(32\) 1.40850 + 0.813199i 0.248990 + 0.143755i
\(33\) 0 0
\(34\) − 0.430037i − 0.0737507i
\(35\) 5.77616 − 8.02237i 0.976349 − 1.35603i
\(36\) 0 0
\(37\) 4.41395 + 7.64519i 0.725650 + 1.25686i 0.958706 + 0.284399i \(0.0917939\pi\)
−0.233057 + 0.972463i \(0.574873\pi\)
\(38\) 0.451713 − 0.782390i 0.0732776 − 0.126920i
\(39\) 0 0
\(40\) −1.76811 + 1.02082i −0.279563 + 0.161406i
\(41\) −6.74011 −1.05263 −0.526314 − 0.850290i \(-0.676427\pi\)
−0.526314 + 0.850290i \(0.676427\pi\)
\(42\) 0 0
\(43\) 0.631237 0.0962627 0.0481313 − 0.998841i \(-0.484673\pi\)
0.0481313 + 0.998841i \(0.484673\pi\)
\(44\) −4.60974 + 2.66144i −0.694945 + 0.401227i
\(45\) 0 0
\(46\) 0.336167 − 0.582258i 0.0495651 − 0.0858493i
\(47\) −3.14006 − 5.43875i −0.458026 − 0.793324i 0.540831 − 0.841131i \(-0.318110\pi\)
−0.998857 + 0.0478078i \(0.984777\pi\)
\(48\) 0 0
\(49\) −4.63910 + 5.24202i −0.662728 + 0.748860i
\(50\) − 1.22983i − 0.173925i
\(51\) 0 0
\(52\) −0.172411 − 0.0995413i −0.0239091 − 0.0138039i
\(53\) −1.05449 − 0.608812i −0.144846 − 0.0836268i 0.425826 − 0.904805i \(-0.359984\pi\)
−0.570671 + 0.821178i \(0.693317\pi\)
\(54\) 0 0
\(55\) − 10.0386i − 1.35361i
\(56\) 1.31818 − 0.593684i 0.176150 − 0.0793344i
\(57\) 0 0
\(58\) −0.603518 − 1.04532i −0.0792458 − 0.137258i
\(59\) 5.43266 − 9.40964i 0.707272 − 1.22503i −0.258593 − 0.965986i \(-0.583259\pi\)
0.965865 − 0.259045i \(-0.0834077\pi\)
\(60\) 0 0
\(61\) 5.52604 − 3.19046i 0.707537 − 0.408497i −0.102611 − 0.994722i \(-0.532720\pi\)
0.810148 + 0.586225i \(0.199386\pi\)
\(62\) −0.229764 −0.0291801
\(63\) 0 0
\(64\) 7.55142 0.943928
\(65\) 0.325156 − 0.187729i 0.0403307 − 0.0232849i
\(66\) 0 0
\(67\) −3.15343 + 5.46190i −0.385253 + 0.667277i −0.991804 − 0.127767i \(-0.959219\pi\)
0.606552 + 0.795044i \(0.292552\pi\)
\(68\) −3.10366 − 5.37569i −0.376374 − 0.651898i
\(69\) 0 0
\(70\) −0.136033 + 1.34997i −0.0162591 + 0.161353i
\(71\) − 2.25070i − 0.267109i −0.991042 − 0.133554i \(-0.957361\pi\)
0.991042 − 0.133554i \(-0.0426390\pi\)
\(72\) 0 0
\(73\) 5.76973 + 3.33116i 0.675296 + 0.389882i 0.798080 − 0.602551i \(-0.205849\pi\)
−0.122784 + 0.992433i \(0.539182\pi\)
\(74\) −1.04932 − 0.605828i −0.121982 − 0.0704261i
\(75\) 0 0
\(76\) − 13.0404i − 1.49584i
\(77\) −0.712691 + 7.07264i −0.0812187 + 0.806002i
\(78\) 0 0
\(79\) −1.79041 − 3.10109i −0.201437 − 0.348900i 0.747554 − 0.664201i \(-0.231228\pi\)
−0.948992 + 0.315301i \(0.897895\pi\)
\(80\) −7.26221 + 12.5785i −0.811940 + 1.40632i
\(81\) 0 0
\(82\) 0.801160 − 0.462550i 0.0884734 − 0.0510801i
\(83\) −7.50252 −0.823508 −0.411754 − 0.911295i \(-0.635084\pi\)
−0.411754 + 0.911295i \(0.635084\pi\)
\(84\) 0 0
\(85\) 11.7066 1.26976
\(86\) −0.0750317 + 0.0433195i −0.00809087 + 0.00467127i
\(87\) 0 0
\(88\) 0.734053 − 1.27142i 0.0782503 − 0.135534i
\(89\) 4.83431 + 8.37327i 0.512436 + 0.887565i 0.999896 + 0.0144199i \(0.00459015\pi\)
−0.487460 + 0.873145i \(0.662077\pi\)
\(90\) 0 0
\(91\) −0.242414 + 0.109179i −0.0254119 + 0.0114450i
\(92\) − 9.70473i − 1.01179i
\(93\) 0 0
\(94\) 0.746485 + 0.430983i 0.0769940 + 0.0444525i
\(95\) 21.2985 + 12.2967i 2.18519 + 1.26162i
\(96\) 0 0
\(97\) − 9.61294i − 0.976046i −0.872831 − 0.488023i \(-0.837718\pi\)
0.872831 − 0.488023i \(-0.162282\pi\)
\(98\) 0.191682 − 0.941455i 0.0193628 − 0.0951013i
\(99\) 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.p.e.80.8 ✓ 32
3.2 odd 2 inner 567.2.p.e.80.9 yes 32
7.5 odd 6 inner 567.2.p.e.404.9 yes 32
9.2 odd 6 567.2.s.g.458.8 32
9.4 even 3 567.2.i.g.269.8 32
9.5 odd 6 567.2.i.g.269.9 32
9.7 even 3 567.2.s.g.458.9 32
21.5 even 6 inner 567.2.p.e.404.8 yes 32
63.5 even 6 567.2.s.g.26.9 32
63.40 odd 6 567.2.s.g.26.8 32
63.47 even 6 567.2.i.g.215.9 32
63.61 odd 6 567.2.i.g.215.8 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
567.2.i.g.215.8 32 63.61 odd 6
567.2.i.g.215.9 32 63.47 even 6
567.2.i.g.269.8 32 9.4 even 3
567.2.i.g.269.9 32 9.5 odd 6
567.2.p.e.80.8 ✓ 32 1.1 even 1 trivial
567.2.p.e.80.9 yes 32 3.2 odd 2 inner
567.2.p.e.404.8 yes 32 21.5 even 6 inner
567.2.p.e.404.9 yes 32 7.5 odd 6 inner
567.2.s.g.26.8 32 63.40 odd 6
567.2.s.g.26.9 32 63.5 even 6
567.2.s.g.458.8 32 9.2 odd 6
567.2.s.g.458.9 32 9.7 even 3