Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [648,2,Mod(217,648)] 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("648.217"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(648, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 648 = 2^{3} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 648.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,-2,0,0] 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: \(5.17430605098\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 24)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 433.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 648.433
Dual form 648.2.i.b.217.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.00000 + 1.73205i) q^{5} +(2.00000 + 3.46410i) q^{11} +(1.00000 - 1.73205i) q^{13} -2.00000 q^{17} -4.00000 q^{19} +(-4.00000 + 6.92820i) q^{23} +(0.500000 + 0.866025i) q^{25} +(3.00000 + 5.19615i) q^{29} +(-4.00000 + 6.92820i) q^{31} +6.00000 q^{37} +(-3.00000 + 5.19615i) q^{41} +(-2.00000 - 3.46410i) q^{43} +(3.50000 - 6.06218i) q^{49} +2.00000 q^{53} -8.00000 q^{55} +(2.00000 - 3.46410i) q^{59} +(1.00000 + 1.73205i) q^{61} +(2.00000 + 3.46410i) q^{65} +(2.00000 - 3.46410i) q^{67} -8.00000 q^{71} +10.0000 q^{73} +(4.00000 + 6.92820i) q^{79} +(-2.00000 - 3.46410i) q^{83} +(2.00000 - 3.46410i) q^{85} +6.00000 q^{89} +(4.00000 - 6.92820i) q^{95} +(-1.00000 - 1.73205i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} + 4 q^{11} + 2 q^{13} - 4 q^{17} - 8 q^{19} - 8 q^{23} + q^{25} + 6 q^{29} - 8 q^{31} + 12 q^{37} - 6 q^{41} - 4 q^{43} + 7 q^{49} + 4 q^{53} - 16 q^{55} + 4 q^{59} + 2 q^{61} + 4 q^{65}+ \cdots - 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/648\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(487\) \(569\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{1}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) −1.00000 + 1.73205i −0.447214 + 0.774597i −0.998203 − 0.0599153i \(-0.980917\pi\)
0.550990 + 0.834512i \(0.314250\pi\)
\(6\) 0 0
\(7\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.00000 + 3.46410i 0.603023 + 1.04447i 0.992361 + 0.123371i \(0.0393705\pi\)
−0.389338 + 0.921095i \(0.627296\pi\)
\(12\) 0 0
\(13\) 1.00000 − 1.73205i 0.277350 − 0.480384i −0.693375 − 0.720577i \(-0.743877\pi\)
0.970725 + 0.240192i \(0.0772105\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −2.00000 −0.485071 −0.242536 − 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) −4.00000 −0.917663 −0.458831 − 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −4.00000 + 6.92820i −0.834058 + 1.44463i 0.0607377 + 0.998154i \(0.480655\pi\)
−0.894795 + 0.446476i \(0.852679\pi\)
\(24\) 0 0
\(25\) 0.500000 + 0.866025i 0.100000 + 0.173205i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 3.00000 + 5.19615i 0.557086 + 0.964901i 0.997738 + 0.0672232i \(0.0214140\pi\)
−0.440652 + 0.897678i \(0.645253\pi\)
\(30\) 0 0
\(31\) −4.00000 + 6.92820i −0.718421 + 1.24434i 0.243204 + 0.969975i \(0.421802\pi\)
−0.961625 + 0.274367i \(0.911532\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 6.00000 0.986394 0.493197 − 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −3.00000 + 5.19615i −0.468521 + 0.811503i −0.999353 − 0.0359748i \(-0.988546\pi\)
0.530831 + 0.847477i \(0.321880\pi\)
\(42\) 0 0
\(43\) −2.00000 − 3.46410i −0.304997 − 0.528271i 0.672264 − 0.740312i \(-0.265322\pi\)
−0.977261 + 0.212041i \(0.931989\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(48\) 0 0
\(49\) 3.50000 − 6.06218i 0.500000 − 0.866025i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 2.00000 0.274721 0.137361 − 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) 0 0
\(55\) −8.00000 −1.07872
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 2.00000 − 3.46410i 0.260378 − 0.450988i −0.705965 − 0.708247i \(-0.749486\pi\)
0.966342 + 0.257260i \(0.0828195\pi\)
\(60\) 0 0
\(61\) 1.00000 + 1.73205i 0.128037 + 0.221766i 0.922916 − 0.385002i \(-0.125799\pi\)
−0.794879 + 0.606768i \(0.792466\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 2.00000 + 3.46410i 0.248069 + 0.429669i
\(66\) 0 0
\(67\) 2.00000 − 3.46410i 0.244339 − 0.423207i −0.717607 − 0.696449i \(-0.754762\pi\)
0.961946 + 0.273241i \(0.0880957\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 − 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) 10.0000 1.17041 0.585206 − 0.810885i \(-0.301014\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 4.00000 + 6.92820i 0.450035 + 0.779484i 0.998388 − 0.0567635i \(-0.0180781\pi\)
−0.548352 + 0.836247i \(0.684745\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −2.00000 − 3.46410i −0.219529 − 0.380235i 0.735135 − 0.677920i \(-0.237119\pi\)
−0.954664 + 0.297686i \(0.903785\pi\)
\(84\) 0 0
\(85\) 2.00000 − 3.46410i 0.216930 − 0.375735i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 − 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 4.00000 − 6.92820i 0.410391 − 0.710819i
\(96\) 0 0
\(97\) −1.00000 − 1.73205i −0.101535 − 0.175863i 0.810782 − 0.585348i \(-0.199042\pi\)
−0.912317 + 0.409484i \(0.865709\pi\)
\(98\) 0 0
\(99\) 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 648.2.i.b.433.1 2
3.2 odd 2 648.2.i.g.433.1 2
4.3 odd 2 1296.2.i.e.433.1 2
9.2 odd 6 648.2.i.g.217.1 2
9.4 even 3 72.2.a.a.1.1 1
9.5 odd 6 24.2.a.a.1.1 ✓ 1
9.7 even 3 inner 648.2.i.b.217.1 2
12.11 even 2 1296.2.i.m.433.1 2
36.7 odd 6 1296.2.i.e.865.1 2
36.11 even 6 1296.2.i.m.865.1 2
36.23 even 6 48.2.a.a.1.1 1
36.31 odd 6 144.2.a.b.1.1 1
45.4 even 6 1800.2.a.m.1.1 1
45.13 odd 12 1800.2.f.c.649.1 2
45.14 odd 6 600.2.a.h.1.1 1
45.22 odd 12 1800.2.f.c.649.2 2
45.23 even 12 600.2.f.e.49.1 2
45.32 even 12 600.2.f.e.49.2 2
63.4 even 3 3528.2.s.j.3313.1 2
63.5 even 6 1176.2.q.a.361.1 2
63.13 odd 6 3528.2.a.d.1.1 1
63.23 odd 6 1176.2.q.i.361.1 2
63.31 odd 6 3528.2.s.y.3313.1 2
63.32 odd 6 1176.2.q.i.961.1 2
63.40 odd 6 3528.2.s.y.361.1 2
63.41 even 6 1176.2.a.i.1.1 1
63.58 even 3 3528.2.s.j.361.1 2
63.59 even 6 1176.2.q.a.961.1 2
72.5 odd 6 192.2.a.d.1.1 1
72.13 even 6 576.2.a.d.1.1 1
72.59 even 6 192.2.a.b.1.1 1
72.67 odd 6 576.2.a.b.1.1 1
99.32 even 6 2904.2.a.c.1.1 1
99.76 odd 6 8712.2.a.u.1.1 1
117.5 even 12 4056.2.c.e.337.2 2
117.77 odd 6 4056.2.a.i.1.1 1
117.86 even 12 4056.2.c.e.337.1 2
144.5 odd 12 768.2.d.e.385.2 2
144.13 even 12 2304.2.d.i.1153.1 2
144.59 even 12 768.2.d.d.385.1 2
144.67 odd 12 2304.2.d.k.1153.1 2
144.77 odd 12 768.2.d.e.385.1 2
144.85 even 12 2304.2.d.i.1153.2 2
144.131 even 12 768.2.d.d.385.2 2
144.139 odd 12 2304.2.d.k.1153.2 2
153.50 odd 6 6936.2.a.p.1.1 1
171.113 even 6 8664.2.a.j.1.1 1
180.23 odd 12 1200.2.f.b.49.2 2
180.59 even 6 1200.2.a.d.1.1 1
180.67 even 12 3600.2.f.r.2449.2 2
180.103 even 12 3600.2.f.r.2449.1 2
180.139 odd 6 3600.2.a.v.1.1 1
180.167 odd 12 1200.2.f.b.49.1 2
252.23 even 6 2352.2.q.l.1537.1 2
252.59 odd 6 2352.2.q.r.961.1 2
252.95 even 6 2352.2.q.l.961.1 2
252.131 odd 6 2352.2.q.r.1537.1 2
252.139 even 6 7056.2.a.q.1.1 1
252.167 odd 6 2352.2.a.i.1.1 1
360.59 even 6 4800.2.a.cc.1.1 1
360.77 even 12 4800.2.f.d.3649.1 2
360.149 odd 6 4800.2.a.q.1.1 1
360.203 odd 12 4800.2.f.bg.3649.1 2
360.293 even 12 4800.2.f.d.3649.2 2
360.347 odd 12 4800.2.f.bg.3649.2 2
396.131 odd 6 5808.2.a.s.1.1 1
468.311 even 6 8112.2.a.be.1.1 1
504.293 even 6 9408.2.a.h.1.1 1
504.419 odd 6 9408.2.a.cc.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
24.2.a.a.1.1 ✓ 1 9.5 odd 6
48.2.a.a.1.1 1 36.23 even 6
72.2.a.a.1.1 1 9.4 even 3
144.2.a.b.1.1 1 36.31 odd 6
192.2.a.b.1.1 1 72.59 even 6
192.2.a.d.1.1 1 72.5 odd 6
576.2.a.b.1.1 1 72.67 odd 6
576.2.a.d.1.1 1 72.13 even 6
600.2.a.h.1.1 1 45.14 odd 6
600.2.f.e.49.1 2 45.23 even 12
600.2.f.e.49.2 2 45.32 even 12
648.2.i.b.217.1 2 9.7 even 3 inner
648.2.i.b.433.1 2 1.1 even 1 trivial
648.2.i.g.217.1 2 9.2 odd 6
648.2.i.g.433.1 2 3.2 odd 2
768.2.d.d.385.1 2 144.59 even 12
768.2.d.d.385.2 2 144.131 even 12
768.2.d.e.385.1 2 144.77 odd 12
768.2.d.e.385.2 2 144.5 odd 12
1176.2.a.i.1.1 1 63.41 even 6
1176.2.q.a.361.1 2 63.5 even 6
1176.2.q.a.961.1 2 63.59 even 6
1176.2.q.i.361.1 2 63.23 odd 6
1176.2.q.i.961.1 2 63.32 odd 6
1200.2.a.d.1.1 1 180.59 even 6
1200.2.f.b.49.1 2 180.167 odd 12
1200.2.f.b.49.2 2 180.23 odd 12
1296.2.i.e.433.1 2 4.3 odd 2
1296.2.i.e.865.1 2 36.7 odd 6
1296.2.i.m.433.1 2 12.11 even 2
1296.2.i.m.865.1 2 36.11 even 6
1800.2.a.m.1.1 1 45.4 even 6
1800.2.f.c.649.1 2 45.13 odd 12
1800.2.f.c.649.2 2 45.22 odd 12
2304.2.d.i.1153.1 2 144.13 even 12
2304.2.d.i.1153.2 2 144.85 even 12
2304.2.d.k.1153.1 2 144.67 odd 12
2304.2.d.k.1153.2 2 144.139 odd 12
2352.2.a.i.1.1 1 252.167 odd 6
2352.2.q.l.961.1 2 252.95 even 6
2352.2.q.l.1537.1 2 252.23 even 6
2352.2.q.r.961.1 2 252.59 odd 6
2352.2.q.r.1537.1 2 252.131 odd 6
2904.2.a.c.1.1 1 99.32 even 6
3528.2.a.d.1.1 1 63.13 odd 6
3528.2.s.j.361.1 2 63.58 even 3
3528.2.s.j.3313.1 2 63.4 even 3
3528.2.s.y.361.1 2 63.40 odd 6
3528.2.s.y.3313.1 2 63.31 odd 6
3600.2.a.v.1.1 1 180.139 odd 6
3600.2.f.r.2449.1 2 180.103 even 12
3600.2.f.r.2449.2 2 180.67 even 12
4056.2.a.i.1.1 1 117.77 odd 6
4056.2.c.e.337.1 2 117.86 even 12
4056.2.c.e.337.2 2 117.5 even 12
4800.2.a.q.1.1 1 360.149 odd 6
4800.2.a.cc.1.1 1 360.59 even 6
4800.2.f.d.3649.1 2 360.77 even 12
4800.2.f.d.3649.2 2 360.293 even 12
4800.2.f.bg.3649.1 2 360.203 odd 12
4800.2.f.bg.3649.2 2 360.347 odd 12
5808.2.a.s.1.1 1 396.131 odd 6
6936.2.a.p.1.1 1 153.50 odd 6
7056.2.a.q.1.1 1 252.139 even 6
8112.2.a.be.1.1 1 468.311 even 6
8664.2.a.j.1.1 1 171.113 even 6
8712.2.a.u.1.1 1 99.76 odd 6
9408.2.a.h.1.1 1 504.293 even 6
9408.2.a.cc.1.1 1 504.419 odd 6