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Properties

Label 4800.2.f.d.3649.2
Level $4800$
Weight $2$
Character 4800.3649
Analytic conductor $38.328$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [4800,2,Mod(3649,4800)] 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("4800.3649"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4800, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 4800 = 2^{6} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4800.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,0,0,-2,0,-8,0,0,0,0,0,0,0,-8,0,0,0,0,0,0,0,0,0,12, 0,16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(31)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(38.3281929702\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) 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: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3649.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 4800.3649
Dual form 4800.2.f.d.3649.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000i q^{3} -1.00000 q^{9} -4.00000 q^{11} +2.00000i q^{13} -2.00000i q^{17} -4.00000 q^{19} -8.00000i q^{23} -1.00000i q^{27} +6.00000 q^{29} +8.00000 q^{31} -4.00000i q^{33} +6.00000i q^{37} -2.00000 q^{39} -6.00000 q^{41} -4.00000i q^{43} +7.00000 q^{49} +2.00000 q^{51} +2.00000i q^{53} -4.00000i q^{57} +4.00000 q^{59} +2.00000 q^{61} -4.00000i q^{67} +8.00000 q^{69} +8.00000 q^{71} +10.0000i q^{73} +8.00000 q^{79} +1.00000 q^{81} +4.00000i q^{83} +6.00000i q^{87} +6.00000 q^{89} +8.00000i q^{93} -2.00000i q^{97} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{9} - 8 q^{11} - 8 q^{19} + 12 q^{29} + 16 q^{31} - 4 q^{39} - 12 q^{41} + 14 q^{49} + 4 q^{51} + 8 q^{59} + 4 q^{61} + 16 q^{69} + 16 q^{71} + 16 q^{79} + 2 q^{81} + 12 q^{89} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4800\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1601\) \(4351\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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 0
\(3\) 1.00000i 0.577350i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 − 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 0 0
\(13\) 2.00000i 0.554700i 0.960769 + 0.277350i \(0.0894562\pi\)
−0.960769 + 0.277350i \(0.910544\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 2.00000i − 0.485071i −0.970143 − 0.242536i \(-0.922021\pi\)
0.970143 − 0.242536i \(-0.0779791\pi\)
\(18\) 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
\(23\) − 8.00000i − 1.66812i −0.551677 − 0.834058i \(-0.686012\pi\)
0.551677 − 0.834058i \(-0.313988\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 1.00000i − 0.192450i
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 − 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) 8.00000 1.43684 0.718421 − 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) 0 0
\(33\) − 4.00000i − 0.696311i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 6.00000i 0.986394i 0.869918 + 0.493197i \(0.164172\pi\)
−0.869918 + 0.493197i \(0.835828\pi\)
\(38\) 0 0
\(39\) −2.00000 −0.320256
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 − 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) − 4.00000i − 0.609994i −0.952353 − 0.304997i \(-0.901344\pi\)
0.952353 − 0.304997i \(-0.0986555\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 7.00000 1.00000
\(50\) 0 0
\(51\) 2.00000 0.280056
\(52\) 0 0
\(53\) 2.00000i 0.274721i 0.990521 + 0.137361i \(0.0438619\pi\)
−0.990521 + 0.137361i \(0.956138\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 4.00000i − 0.529813i
\(58\) 0 0
\(59\) 4.00000 0.520756 0.260378 − 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 − 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 4.00000i − 0.488678i −0.969690 − 0.244339i \(-0.921429\pi\)
0.969690 − 0.244339i \(-0.0785709\pi\)
\(68\) 0 0
\(69\) 8.00000 0.963087
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 − 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 0 0
\(73\) 10.0000i 1.17041i 0.810885 + 0.585206i \(0.198986\pi\)
−0.810885 + 0.585206i \(0.801014\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 − 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 4.00000i 0.439057i 0.975606 + 0.219529i \(0.0704519\pi\)
−0.975606 + 0.219529i \(0.929548\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 6.00000i 0.643268i
\(88\) 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
\(93\) 8.00000i 0.829561i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 2.00000i − 0.203069i −0.994832 − 0.101535i \(-0.967625\pi\)
0.994832 − 0.101535i \(-0.0323753\pi\)
\(98\) 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 4800.2.f.d.3649.2 2
4.3 odd 2 4800.2.f.bg.3649.1 2
5.2 odd 4 192.2.a.d.1.1 1
5.3 odd 4 4800.2.a.q.1.1 1
5.4 even 2 inner 4800.2.f.d.3649.1 2
8.3 odd 2 1200.2.f.b.49.2 2
8.5 even 2 600.2.f.e.49.1 2
15.2 even 4 576.2.a.d.1.1 1
20.3 even 4 4800.2.a.cc.1.1 1
20.7 even 4 192.2.a.b.1.1 1
20.19 odd 2 4800.2.f.bg.3649.2 2
24.5 odd 2 1800.2.f.c.649.1 2
24.11 even 2 3600.2.f.r.2449.1 2
35.27 even 4 9408.2.a.h.1.1 1
40.3 even 4 1200.2.a.d.1.1 1
40.13 odd 4 600.2.a.h.1.1 1
40.19 odd 2 1200.2.f.b.49.1 2
40.27 even 4 48.2.a.a.1.1 1
40.29 even 2 600.2.f.e.49.2 2
40.37 odd 4 24.2.a.a.1.1 ✓ 1
60.47 odd 4 576.2.a.b.1.1 1
80.27 even 4 768.2.d.d.385.2 2
80.37 odd 4 768.2.d.e.385.1 2
80.67 even 4 768.2.d.d.385.1 2
80.77 odd 4 768.2.d.e.385.2 2
120.29 odd 2 1800.2.f.c.649.2 2
120.53 even 4 1800.2.a.m.1.1 1
120.59 even 2 3600.2.f.r.2449.2 2
120.77 even 4 72.2.a.a.1.1 1
120.83 odd 4 3600.2.a.v.1.1 1
120.107 odd 4 144.2.a.b.1.1 1
140.27 odd 4 9408.2.a.cc.1.1 1
240.77 even 4 2304.2.d.i.1153.2 2
240.107 odd 4 2304.2.d.k.1153.1 2
240.197 even 4 2304.2.d.i.1153.1 2
240.227 odd 4 2304.2.d.k.1153.2 2
280.27 odd 4 2352.2.a.i.1.1 1
280.37 odd 12 1176.2.q.i.361.1 2
280.67 even 12 2352.2.q.l.961.1 2
280.107 even 12 2352.2.q.l.1537.1 2
280.117 even 12 1176.2.q.a.361.1 2
280.157 even 12 1176.2.q.a.961.1 2
280.187 odd 12 2352.2.q.r.1537.1 2
280.227 odd 12 2352.2.q.r.961.1 2
280.237 even 4 1176.2.a.i.1.1 1
280.277 odd 12 1176.2.q.i.961.1 2
360.67 even 12 1296.2.i.m.865.1 2
360.77 even 12 648.2.i.b.217.1 2
360.157 odd 12 648.2.i.g.217.1 2
360.187 even 12 1296.2.i.m.433.1 2
360.227 odd 12 1296.2.i.e.433.1 2
360.277 odd 12 648.2.i.g.433.1 2
360.317 even 12 648.2.i.b.433.1 2
360.347 odd 12 1296.2.i.e.865.1 2
440.197 even 4 2904.2.a.c.1.1 1
440.307 odd 4 5808.2.a.s.1.1 1
520.77 odd 4 4056.2.a.i.1.1 1
520.317 even 4 4056.2.c.e.337.2 2
520.437 even 4 4056.2.c.e.337.1 2
520.467 even 4 8112.2.a.be.1.1 1
680.237 odd 4 6936.2.a.p.1.1 1
760.37 even 4 8664.2.a.j.1.1 1
840.317 even 12 3528.2.s.j.361.1 2
840.437 odd 12 3528.2.s.y.3313.1 2
840.557 even 12 3528.2.s.j.3313.1 2
840.587 even 4 7056.2.a.q.1.1 1
840.677 odd 12 3528.2.s.y.361.1 2
840.797 odd 4 3528.2.a.d.1.1 1
1320.197 odd 4 8712.2.a.u.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
24.2.a.a.1.1 ✓ 1 40.37 odd 4
48.2.a.a.1.1 1 40.27 even 4
72.2.a.a.1.1 1 120.77 even 4
144.2.a.b.1.1 1 120.107 odd 4
192.2.a.b.1.1 1 20.7 even 4
192.2.a.d.1.1 1 5.2 odd 4
576.2.a.b.1.1 1 60.47 odd 4
576.2.a.d.1.1 1 15.2 even 4
600.2.a.h.1.1 1 40.13 odd 4
600.2.f.e.49.1 2 8.5 even 2
600.2.f.e.49.2 2 40.29 even 2
648.2.i.b.217.1 2 360.77 even 12
648.2.i.b.433.1 2 360.317 even 12
648.2.i.g.217.1 2 360.157 odd 12
648.2.i.g.433.1 2 360.277 odd 12
768.2.d.d.385.1 2 80.67 even 4
768.2.d.d.385.2 2 80.27 even 4
768.2.d.e.385.1 2 80.37 odd 4
768.2.d.e.385.2 2 80.77 odd 4
1176.2.a.i.1.1 1 280.237 even 4
1176.2.q.a.361.1 2 280.117 even 12
1176.2.q.a.961.1 2 280.157 even 12
1176.2.q.i.361.1 2 280.37 odd 12
1176.2.q.i.961.1 2 280.277 odd 12
1200.2.a.d.1.1 1 40.3 even 4
1200.2.f.b.49.1 2 40.19 odd 2
1200.2.f.b.49.2 2 8.3 odd 2
1296.2.i.e.433.1 2 360.227 odd 12
1296.2.i.e.865.1 2 360.347 odd 12
1296.2.i.m.433.1 2 360.187 even 12
1296.2.i.m.865.1 2 360.67 even 12
1800.2.a.m.1.1 1 120.53 even 4
1800.2.f.c.649.1 2 24.5 odd 2
1800.2.f.c.649.2 2 120.29 odd 2
2304.2.d.i.1153.1 2 240.197 even 4
2304.2.d.i.1153.2 2 240.77 even 4
2304.2.d.k.1153.1 2 240.107 odd 4
2304.2.d.k.1153.2 2 240.227 odd 4
2352.2.a.i.1.1 1 280.27 odd 4
2352.2.q.l.961.1 2 280.67 even 12
2352.2.q.l.1537.1 2 280.107 even 12
2352.2.q.r.961.1 2 280.227 odd 12
2352.2.q.r.1537.1 2 280.187 odd 12
2904.2.a.c.1.1 1 440.197 even 4
3528.2.a.d.1.1 1 840.797 odd 4
3528.2.s.j.361.1 2 840.317 even 12
3528.2.s.j.3313.1 2 840.557 even 12
3528.2.s.y.361.1 2 840.677 odd 12
3528.2.s.y.3313.1 2 840.437 odd 12
3600.2.a.v.1.1 1 120.83 odd 4
3600.2.f.r.2449.1 2 24.11 even 2
3600.2.f.r.2449.2 2 120.59 even 2
4056.2.a.i.1.1 1 520.77 odd 4
4056.2.c.e.337.1 2 520.437 even 4
4056.2.c.e.337.2 2 520.317 even 4
4800.2.a.q.1.1 1 5.3 odd 4
4800.2.a.cc.1.1 1 20.3 even 4
4800.2.f.d.3649.1 2 5.4 even 2 inner
4800.2.f.d.3649.2 2 1.1 even 1 trivial
4800.2.f.bg.3649.1 2 4.3 odd 2
4800.2.f.bg.3649.2 2 20.19 odd 2
5808.2.a.s.1.1 1 440.307 odd 4
6936.2.a.p.1.1 1 680.237 odd 4
7056.2.a.q.1.1 1 840.587 even 4
8112.2.a.be.1.1 1 520.467 even 4
8664.2.a.j.1.1 1 760.37 even 4
8712.2.a.u.1.1 1 1320.197 odd 4
9408.2.a.h.1.1 1 35.27 even 4
9408.2.a.cc.1.1 1 140.27 odd 4