Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2352,2,Mod(961,2352)] 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("2352.961"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2352, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2352.q (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-1,0,2,0,0,0,-1,0,4,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(18.7808145554\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 24)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 1537.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 2352.1537
Dual form 2352.2.q.l.961.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.500000 + 0.866025i) q^{3} +(1.00000 + 1.73205i) q^{5} +(-0.500000 - 0.866025i) q^{9} +(2.00000 - 3.46410i) q^{11} -2.00000 q^{13} -2.00000 q^{15} +(-1.00000 + 1.73205i) q^{17} +(-2.00000 - 3.46410i) q^{19} +(-4.00000 - 6.92820i) q^{23} +(0.500000 - 0.866025i) q^{25} +1.00000 q^{27} +6.00000 q^{29} +(4.00000 - 6.92820i) q^{31} +(2.00000 + 3.46410i) q^{33} +(-3.00000 - 5.19615i) q^{37} +(1.00000 - 1.73205i) q^{39} -6.00000 q^{41} -4.00000 q^{43} +(1.00000 - 1.73205i) q^{45} +(-1.00000 - 1.73205i) q^{51} +(1.00000 - 1.73205i) q^{53} +8.00000 q^{55} +4.00000 q^{57} +(2.00000 - 3.46410i) q^{59} +(1.00000 + 1.73205i) q^{61} +(-2.00000 - 3.46410i) q^{65} +(-2.00000 + 3.46410i) q^{67} +8.00000 q^{69} -8.00000 q^{71} +(-5.00000 + 8.66025i) q^{73} +(0.500000 + 0.866025i) q^{75} +(-4.00000 - 6.92820i) q^{79} +(-0.500000 + 0.866025i) q^{81} +4.00000 q^{83} -4.00000 q^{85} +(-3.00000 + 5.19615i) q^{87} +(3.00000 + 5.19615i) q^{89} +(4.00000 + 6.92820i) q^{93} +(4.00000 - 6.92820i) q^{95} +2.00000 q^{97} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} + 2 q^{5} - q^{9} + 4 q^{11} - 4 q^{13} - 4 q^{15} - 2 q^{17} - 4 q^{19} - 8 q^{23} + q^{25} + 2 q^{27} + 12 q^{29} + 8 q^{31} + 4 q^{33} - 6 q^{37} + 2 q^{39} - 12 q^{41} - 8 q^{43} + 2 q^{45}+ \cdots - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1471\) \(1765\) \(2257\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −0.500000 + 0.866025i −0.288675 + 0.500000i
\(4\) 0 0
\(5\) 1.00000 + 1.73205i 0.447214 + 0.774597i 0.998203 − 0.0599153i \(-0.0190830\pi\)
−0.550990 + 0.834512i \(0.685750\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −0.500000 − 0.866025i −0.166667 − 0.288675i
\(10\) 0 0
\(11\) 2.00000 − 3.46410i 0.603023 − 1.04447i −0.389338 − 0.921095i \(-0.627296\pi\)
0.992361 − 0.123371i \(-0.0393705\pi\)
\(12\) 0 0
\(13\) −2.00000 −0.554700 −0.277350 − 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) −2.00000 −0.516398
\(16\) 0 0
\(17\) −1.00000 + 1.73205i −0.242536 + 0.420084i −0.961436 − 0.275029i \(-0.911312\pi\)
0.718900 + 0.695113i \(0.244646\pi\)
\(18\) 0 0
\(19\) −2.00000 − 3.46410i −0.458831 − 0.794719i 0.540068 − 0.841621i \(-0.318398\pi\)
−0.998899 + 0.0469020i \(0.985065\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −4.00000 − 6.92820i −0.834058 − 1.44463i −0.894795 − 0.446476i \(-0.852679\pi\)
0.0607377 − 0.998154i \(-0.480655\pi\)
\(24\) 0 0
\(25\) 0.500000 − 0.866025i 0.100000 − 0.173205i
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 − 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) 4.00000 − 6.92820i 0.718421 − 1.24434i −0.243204 − 0.969975i \(-0.578198\pi\)
0.961625 − 0.274367i \(-0.0884683\pi\)
\(32\) 0 0
\(33\) 2.00000 + 3.46410i 0.348155 + 0.603023i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −3.00000 − 5.19615i −0.493197 − 0.854242i 0.506772 − 0.862080i \(-0.330838\pi\)
−0.999969 + 0.00783774i \(0.997505\pi\)
\(38\) 0 0
\(39\) 1.00000 − 1.73205i 0.160128 − 0.277350i
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 − 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 − 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) 1.00000 − 1.73205i 0.149071 − 0.258199i
\(46\) 0 0
\(47\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −1.00000 − 1.73205i −0.140028 − 0.242536i
\(52\) 0 0
\(53\) 1.00000 − 1.73205i 0.137361 − 0.237915i −0.789136 − 0.614218i \(-0.789471\pi\)
0.926497 + 0.376303i \(0.122805\pi\)
\(54\) 0 0
\(55\) 8.00000 1.07872
\(56\) 0 0
\(57\) 4.00000 0.529813
\(58\) 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 −0.961946 − 0.273241i \(-0.911904\pi\)
0.717607 + 0.696449i \(0.245238\pi\)
\(68\) 0 0
\(69\) 8.00000 0.963087
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 − 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) −5.00000 + 8.66025i −0.585206 + 1.01361i 0.409644 + 0.912245i \(0.365653\pi\)
−0.994850 + 0.101361i \(0.967680\pi\)
\(74\) 0 0
\(75\) 0.500000 + 0.866025i 0.0577350 + 0.100000i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −4.00000 − 6.92820i −0.450035 − 0.779484i 0.548352 − 0.836247i \(-0.315255\pi\)
−0.998388 + 0.0567635i \(0.981922\pi\)
\(80\) 0 0
\(81\) −0.500000 + 0.866025i −0.0555556 + 0.0962250i
\(82\) 0 0
\(83\) 4.00000 0.439057 0.219529 − 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 0 0
\(85\) −4.00000 −0.433861
\(86\) 0 0
\(87\) −3.00000 + 5.19615i −0.321634 + 0.557086i
\(88\) 0 0
\(89\) 3.00000 + 5.19615i 0.317999 + 0.550791i 0.980071 − 0.198650i \(-0.0636557\pi\)
−0.662071 + 0.749441i \(0.730322\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 4.00000 + 6.92820i 0.414781 + 0.718421i
\(94\) 0 0
\(95\) 4.00000 − 6.92820i 0.410391 − 0.710819i
\(96\) 0 0
\(97\) 2.00000 0.203069 0.101535 − 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 0 0
\(99\) −4.00000 −0.402015
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 2352.2.q.l.1537.1 2
4.3 odd 2 1176.2.q.i.361.1 2
7.2 even 3 inner 2352.2.q.l.961.1 2
7.3 odd 6 2352.2.a.i.1.1 1
7.4 even 3 48.2.a.a.1.1 1
7.5 odd 6 2352.2.q.r.961.1 2
7.6 odd 2 2352.2.q.r.1537.1 2
12.11 even 2 3528.2.s.j.361.1 2
21.11 odd 6 144.2.a.b.1.1 1
21.17 even 6 7056.2.a.q.1.1 1
28.3 even 6 1176.2.a.i.1.1 1
28.11 odd 6 24.2.a.a.1.1 ✓ 1
28.19 even 6 1176.2.q.a.961.1 2
28.23 odd 6 1176.2.q.i.961.1 2
28.27 even 2 1176.2.q.a.361.1 2
35.4 even 6 1200.2.a.d.1.1 1
35.18 odd 12 1200.2.f.b.49.2 2
35.32 odd 12 1200.2.f.b.49.1 2
56.3 even 6 9408.2.a.h.1.1 1
56.11 odd 6 192.2.a.d.1.1 1
56.45 odd 6 9408.2.a.cc.1.1 1
56.53 even 6 192.2.a.b.1.1 1
63.4 even 3 1296.2.i.m.865.1 2
63.11 odd 6 1296.2.i.e.433.1 2
63.25 even 3 1296.2.i.m.433.1 2
63.32 odd 6 1296.2.i.e.865.1 2
77.32 odd 6 5808.2.a.s.1.1 1
84.11 even 6 72.2.a.a.1.1 1
84.23 even 6 3528.2.s.j.3313.1 2
84.47 odd 6 3528.2.s.y.3313.1 2
84.59 odd 6 3528.2.a.d.1.1 1
84.83 odd 2 3528.2.s.y.361.1 2
91.25 even 6 8112.2.a.be.1.1 1
105.32 even 12 3600.2.f.r.2449.2 2
105.53 even 12 3600.2.f.r.2449.1 2
105.74 odd 6 3600.2.a.v.1.1 1
112.11 odd 12 768.2.d.e.385.2 2
112.53 even 12 768.2.d.d.385.1 2
112.67 odd 12 768.2.d.e.385.1 2
112.109 even 12 768.2.d.d.385.2 2
140.39 odd 6 600.2.a.h.1.1 1
140.67 even 12 600.2.f.e.49.2 2
140.123 even 12 600.2.f.e.49.1 2
168.11 even 6 576.2.a.d.1.1 1
168.53 odd 6 576.2.a.b.1.1 1
252.11 even 6 648.2.i.b.433.1 2
252.67 odd 6 648.2.i.g.217.1 2
252.95 even 6 648.2.i.b.217.1 2
252.151 odd 6 648.2.i.g.433.1 2
280.53 odd 12 4800.2.f.bg.3649.1 2
280.67 even 12 4800.2.f.d.3649.1 2
280.109 even 6 4800.2.a.cc.1.1 1
280.123 even 12 4800.2.f.d.3649.2 2
280.179 odd 6 4800.2.a.q.1.1 1
280.277 odd 12 4800.2.f.bg.3649.2 2
308.263 even 6 2904.2.a.c.1.1 1
336.11 even 12 2304.2.d.i.1153.2 2
336.53 odd 12 2304.2.d.k.1153.2 2
336.179 even 12 2304.2.d.i.1153.1 2
336.221 odd 12 2304.2.d.k.1153.1 2
364.151 even 12 4056.2.c.e.337.1 2
364.207 odd 6 4056.2.a.i.1.1 1
364.291 even 12 4056.2.c.e.337.2 2
420.179 even 6 1800.2.a.m.1.1 1
420.263 odd 12 1800.2.f.c.649.1 2
420.347 odd 12 1800.2.f.c.649.2 2
476.67 odd 6 6936.2.a.p.1.1 1
532.151 even 6 8664.2.a.j.1.1 1
924.263 odd 6 8712.2.a.u.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
24.2.a.a.1.1 ✓ 1 28.11 odd 6
48.2.a.a.1.1 1 7.4 even 3
72.2.a.a.1.1 1 84.11 even 6
144.2.a.b.1.1 1 21.11 odd 6
192.2.a.b.1.1 1 56.53 even 6
192.2.a.d.1.1 1 56.11 odd 6
576.2.a.b.1.1 1 168.53 odd 6
576.2.a.d.1.1 1 168.11 even 6
600.2.a.h.1.1 1 140.39 odd 6
600.2.f.e.49.1 2 140.123 even 12
600.2.f.e.49.2 2 140.67 even 12
648.2.i.b.217.1 2 252.95 even 6
648.2.i.b.433.1 2 252.11 even 6
648.2.i.g.217.1 2 252.67 odd 6
648.2.i.g.433.1 2 252.151 odd 6
768.2.d.d.385.1 2 112.53 even 12
768.2.d.d.385.2 2 112.109 even 12
768.2.d.e.385.1 2 112.67 odd 12
768.2.d.e.385.2 2 112.11 odd 12
1176.2.a.i.1.1 1 28.3 even 6
1176.2.q.a.361.1 2 28.27 even 2
1176.2.q.a.961.1 2 28.19 even 6
1176.2.q.i.361.1 2 4.3 odd 2
1176.2.q.i.961.1 2 28.23 odd 6
1200.2.a.d.1.1 1 35.4 even 6
1200.2.f.b.49.1 2 35.32 odd 12
1200.2.f.b.49.2 2 35.18 odd 12
1296.2.i.e.433.1 2 63.11 odd 6
1296.2.i.e.865.1 2 63.32 odd 6
1296.2.i.m.433.1 2 63.25 even 3
1296.2.i.m.865.1 2 63.4 even 3
1800.2.a.m.1.1 1 420.179 even 6
1800.2.f.c.649.1 2 420.263 odd 12
1800.2.f.c.649.2 2 420.347 odd 12
2304.2.d.i.1153.1 2 336.179 even 12
2304.2.d.i.1153.2 2 336.11 even 12
2304.2.d.k.1153.1 2 336.221 odd 12
2304.2.d.k.1153.2 2 336.53 odd 12
2352.2.a.i.1.1 1 7.3 odd 6
2352.2.q.l.961.1 2 7.2 even 3 inner
2352.2.q.l.1537.1 2 1.1 even 1 trivial
2352.2.q.r.961.1 2 7.5 odd 6
2352.2.q.r.1537.1 2 7.6 odd 2
2904.2.a.c.1.1 1 308.263 even 6
3528.2.a.d.1.1 1 84.59 odd 6
3528.2.s.j.361.1 2 12.11 even 2
3528.2.s.j.3313.1 2 84.23 even 6
3528.2.s.y.361.1 2 84.83 odd 2
3528.2.s.y.3313.1 2 84.47 odd 6
3600.2.a.v.1.1 1 105.74 odd 6
3600.2.f.r.2449.1 2 105.53 even 12
3600.2.f.r.2449.2 2 105.32 even 12
4056.2.a.i.1.1 1 364.207 odd 6
4056.2.c.e.337.1 2 364.151 even 12
4056.2.c.e.337.2 2 364.291 even 12
4800.2.a.q.1.1 1 280.179 odd 6
4800.2.a.cc.1.1 1 280.109 even 6
4800.2.f.d.3649.1 2 280.67 even 12
4800.2.f.d.3649.2 2 280.123 even 12
4800.2.f.bg.3649.1 2 280.53 odd 12
4800.2.f.bg.3649.2 2 280.277 odd 12
5808.2.a.s.1.1 1 77.32 odd 6
6936.2.a.p.1.1 1 476.67 odd 6
7056.2.a.q.1.1 1 21.17 even 6
8112.2.a.be.1.1 1 91.25 even 6
8664.2.a.j.1.1 1 532.151 even 6
8712.2.a.u.1.1 1 924.263 odd 6
9408.2.a.h.1.1 1 56.3 even 6
9408.2.a.cc.1.1 1 56.45 odd 6