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Properties

Label 3840.2.d.o.2689.1
Level $3840$
Weight $2$
Character 3840.2689
Analytic conductor $30.663$
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] = [3840,2,Mod(2689,3840)] 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("3840.2689"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3840, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3840 = 2^{8} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3840.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-2,0,4,0,0,0,2,0,0,0,0,0,-4,0,0,0,0,0,0,0,0,0,6,0,-2,0,0, 0,8,0,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(35)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(30.6625543762\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 60)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2689.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 3840.2689
Dual form 3840.2.d.o.2689.2

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} +(2.00000 - 1.00000i) q^{5} +4.00000i q^{7} +1.00000 q^{9} +4.00000i q^{11} +(-2.00000 + 1.00000i) q^{15} -4.00000i q^{17} -4.00000i q^{21} -4.00000i q^{23} +(3.00000 - 4.00000i) q^{25} -1.00000 q^{27} +6.00000i q^{29} +4.00000 q^{31} -4.00000i q^{33} +(4.00000 + 8.00000i) q^{35} +8.00000 q^{37} +10.0000 q^{41} +4.00000 q^{43} +(2.00000 - 1.00000i) q^{45} +4.00000i q^{47} -9.00000 q^{49} +4.00000i q^{51} -12.0000 q^{53} +(4.00000 + 8.00000i) q^{55} +4.00000i q^{59} +2.00000i q^{61} +4.00000i q^{63} -4.00000 q^{67} +4.00000i q^{69} +8.00000i q^{73} +(-3.00000 + 4.00000i) q^{75} -16.0000 q^{77} +12.0000 q^{79} +1.00000 q^{81} -4.00000 q^{83} +(-4.00000 - 8.00000i) q^{85} -6.00000i q^{87} -10.0000 q^{89} -4.00000 q^{93} -8.00000i q^{97} +4.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 4 q^{5} + 2 q^{9} - 4 q^{15} + 6 q^{25} - 2 q^{27} + 8 q^{31} + 8 q^{35} + 16 q^{37} + 20 q^{41} + 8 q^{43} + 4 q^{45} - 18 q^{49} - 24 q^{53} + 8 q^{55} - 8 q^{67} - 6 q^{75} - 32 q^{77}+ \cdots - 8 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3840\mathbb{Z}\right)^\times\).

\(n\) \(511\) \(1537\) \(2561\) \(2821\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(-1\)

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
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 2.00000 − 1.00000i 0.894427 − 0.447214i
\(6\) 0 0
\(7\) 4.00000i 1.51186i 0.654654 + 0.755929i \(0.272814\pi\)
−0.654654 + 0.755929i \(0.727186\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 4.00000i 1.20605i 0.797724 + 0.603023i \(0.206037\pi\)
−0.797724 + 0.603023i \(0.793963\pi\)
\(12\) 0 0
\(13\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) −2.00000 + 1.00000i −0.516398 + 0.258199i
\(16\) 0 0
\(17\) − 4.00000i − 0.970143i −0.874475 − 0.485071i \(-0.838794\pi\)
0.874475 − 0.485071i \(-0.161206\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) − 4.00000i − 0.872872i
\(22\) 0 0
\(23\) − 4.00000i − 0.834058i −0.908893 − 0.417029i \(-0.863071\pi\)
0.908893 − 0.417029i \(-0.136929\pi\)
\(24\) 0 0
\(25\) 3.00000 − 4.00000i 0.600000 − 0.800000i
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 6.00000i 1.11417i 0.830455 + 0.557086i \(0.188081\pi\)
−0.830455 + 0.557086i \(0.811919\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 − 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 0 0
\(33\) − 4.00000i − 0.696311i
\(34\) 0 0
\(35\) 4.00000 + 8.00000i 0.676123 + 1.35225i
\(36\) 0 0
\(37\) 8.00000 1.31519 0.657596 − 0.753371i \(-0.271573\pi\)
0.657596 + 0.753371i \(0.271573\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 10.0000 1.56174 0.780869 − 0.624695i \(-0.214777\pi\)
0.780869 + 0.624695i \(0.214777\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 − 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 0 0
\(45\) 2.00000 − 1.00000i 0.298142 − 0.149071i
\(46\) 0 0
\(47\) 4.00000i 0.583460i 0.956501 + 0.291730i \(0.0942309\pi\)
−0.956501 + 0.291730i \(0.905769\pi\)
\(48\) 0 0
\(49\) −9.00000 −1.28571
\(50\) 0 0
\(51\) 4.00000i 0.560112i
\(52\) 0 0
\(53\) −12.0000 −1.64833 −0.824163 − 0.566352i \(-0.808354\pi\)
−0.824163 + 0.566352i \(0.808354\pi\)
\(54\) 0 0
\(55\) 4.00000 + 8.00000i 0.539360 + 1.07872i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 4.00000i 0.520756i 0.965507 + 0.260378i \(0.0838471\pi\)
−0.965507 + 0.260378i \(0.916153\pi\)
\(60\) 0 0
\(61\) 2.00000i 0.256074i 0.991769 + 0.128037i \(0.0408676\pi\)
−0.991769 + 0.128037i \(0.959132\pi\)
\(62\) 0 0
\(63\) 4.00000i 0.503953i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 − 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 0 0
\(69\) 4.00000i 0.481543i
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 8.00000i 0.936329i 0.883641 + 0.468165i \(0.155085\pi\)
−0.883641 + 0.468165i \(0.844915\pi\)
\(74\) 0 0
\(75\) −3.00000 + 4.00000i −0.346410 + 0.461880i
\(76\) 0 0
\(77\) −16.0000 −1.82337
\(78\) 0 0
\(79\) 12.0000 1.35011 0.675053 − 0.737769i \(-0.264121\pi\)
0.675053 + 0.737769i \(0.264121\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −4.00000 −0.439057 −0.219529 − 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) −4.00000 − 8.00000i −0.433861 − 0.867722i
\(86\) 0 0
\(87\) − 6.00000i − 0.643268i
\(88\) 0 0
\(89\) −10.0000 −1.06000 −0.529999 − 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −4.00000 −0.414781
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 8.00000i − 0.812277i −0.913812 − 0.406138i \(-0.866875\pi\)
0.913812 − 0.406138i \(-0.133125\pi\)
\(98\) 0 0
\(99\) 4.00000i 0.402015i
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 3840.2.d.o.2689.1 2
4.3 odd 2 3840.2.d.be.2689.1 2
5.4 even 2 3840.2.d.r.2689.1 2
8.3 odd 2 3840.2.d.b.2689.2 2
8.5 even 2 3840.2.d.r.2689.2 2
16.3 odd 4 960.2.f.c.769.2 2
16.5 even 4 60.2.d.a.49.2 yes 2
16.11 odd 4 240.2.f.b.49.1 2
16.13 even 4 960.2.f.f.769.1 2
20.19 odd 2 3840.2.d.b.2689.1 2
40.19 odd 2 3840.2.d.be.2689.2 2
40.29 even 2 inner 3840.2.d.o.2689.2 2
48.5 odd 4 180.2.d.a.109.1 2
48.11 even 4 720.2.f.c.289.1 2
48.29 odd 4 2880.2.f.l.1729.2 2
48.35 even 4 2880.2.f.p.1729.2 2
80.3 even 4 4800.2.a.bf.1.1 1
80.13 odd 4 4800.2.a.bn.1.1 1
80.19 odd 4 960.2.f.c.769.1 2
80.27 even 4 1200.2.a.a.1.1 1
80.29 even 4 960.2.f.f.769.2 2
80.37 odd 4 300.2.a.d.1.1 1
80.43 even 4 1200.2.a.s.1.1 1
80.53 odd 4 300.2.a.a.1.1 1
80.59 odd 4 240.2.f.b.49.2 2
80.67 even 4 4800.2.a.bk.1.1 1
80.69 even 4 60.2.d.a.49.1 ✓ 2
80.77 odd 4 4800.2.a.bj.1.1 1
112.5 odd 12 2940.2.bb.e.949.2 4
112.37 even 12 2940.2.bb.d.949.1 4
112.53 even 12 2940.2.bb.d.1549.2 4
112.69 odd 4 2940.2.k.c.589.1 2
112.101 odd 12 2940.2.bb.e.1549.1 4
144.5 odd 12 1620.2.r.d.1189.1 4
144.85 even 12 1620.2.r.c.1189.2 4
144.101 odd 12 1620.2.r.d.109.2 4
144.133 even 12 1620.2.r.c.109.1 4
240.29 odd 4 2880.2.f.l.1729.1 2
240.53 even 4 900.2.a.a.1.1 1
240.59 even 4 720.2.f.c.289.2 2
240.107 odd 4 3600.2.a.d.1.1 1
240.149 odd 4 180.2.d.a.109.2 2
240.179 even 4 2880.2.f.p.1729.1 2
240.197 even 4 900.2.a.h.1.1 1
240.203 odd 4 3600.2.a.bm.1.1 1
560.69 odd 4 2940.2.k.c.589.2 2
560.149 even 12 2940.2.bb.d.949.2 4
560.229 odd 12 2940.2.bb.e.949.1 4
560.389 even 12 2940.2.bb.d.1549.1 4
560.549 odd 12 2940.2.bb.e.1549.2 4
720.149 odd 12 1620.2.r.d.1189.2 4
720.229 even 12 1620.2.r.c.1189.1 4
720.389 odd 12 1620.2.r.d.109.1 4
720.709 even 12 1620.2.r.c.109.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
60.2.d.a.49.1 ✓ 2 80.69 even 4
60.2.d.a.49.2 yes 2 16.5 even 4
180.2.d.a.109.1 2 48.5 odd 4
180.2.d.a.109.2 2 240.149 odd 4
240.2.f.b.49.1 2 16.11 odd 4
240.2.f.b.49.2 2 80.59 odd 4
300.2.a.a.1.1 1 80.53 odd 4
300.2.a.d.1.1 1 80.37 odd 4
720.2.f.c.289.1 2 48.11 even 4
720.2.f.c.289.2 2 240.59 even 4
900.2.a.a.1.1 1 240.53 even 4
900.2.a.h.1.1 1 240.197 even 4
960.2.f.c.769.1 2 80.19 odd 4
960.2.f.c.769.2 2 16.3 odd 4
960.2.f.f.769.1 2 16.13 even 4
960.2.f.f.769.2 2 80.29 even 4
1200.2.a.a.1.1 1 80.27 even 4
1200.2.a.s.1.1 1 80.43 even 4
1620.2.r.c.109.1 4 144.133 even 12
1620.2.r.c.109.2 4 720.709 even 12
1620.2.r.c.1189.1 4 720.229 even 12
1620.2.r.c.1189.2 4 144.85 even 12
1620.2.r.d.109.1 4 720.389 odd 12
1620.2.r.d.109.2 4 144.101 odd 12
1620.2.r.d.1189.1 4 144.5 odd 12
1620.2.r.d.1189.2 4 720.149 odd 12
2880.2.f.l.1729.1 2 240.29 odd 4
2880.2.f.l.1729.2 2 48.29 odd 4
2880.2.f.p.1729.1 2 240.179 even 4
2880.2.f.p.1729.2 2 48.35 even 4
2940.2.k.c.589.1 2 112.69 odd 4
2940.2.k.c.589.2 2 560.69 odd 4
2940.2.bb.d.949.1 4 112.37 even 12
2940.2.bb.d.949.2 4 560.149 even 12
2940.2.bb.d.1549.1 4 560.389 even 12
2940.2.bb.d.1549.2 4 112.53 even 12
2940.2.bb.e.949.1 4 560.229 odd 12
2940.2.bb.e.949.2 4 112.5 odd 12
2940.2.bb.e.1549.1 4 112.101 odd 12
2940.2.bb.e.1549.2 4 560.549 odd 12
3600.2.a.d.1.1 1 240.107 odd 4
3600.2.a.bm.1.1 1 240.203 odd 4
3840.2.d.b.2689.1 2 20.19 odd 2
3840.2.d.b.2689.2 2 8.3 odd 2
3840.2.d.o.2689.1 2 1.1 even 1 trivial
3840.2.d.o.2689.2 2 40.29 even 2 inner
3840.2.d.r.2689.1 2 5.4 even 2
3840.2.d.r.2689.2 2 8.5 even 2
3840.2.d.be.2689.1 2 4.3 odd 2
3840.2.d.be.2689.2 2 40.19 odd 2
4800.2.a.bf.1.1 1 80.3 even 4
4800.2.a.bj.1.1 1 80.77 odd 4
4800.2.a.bk.1.1 1 80.67 even 4
4800.2.a.bn.1.1 1 80.13 odd 4