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Properties

Label 4056.2.c.e.337.2
Level $4056$
Weight $2$
Character 4056.337
Analytic conductor $32.387$
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] = [4056,2,Mod(337,4056)] 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("4056.337"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4056, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 4056 = 2^{3} \cdot 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4056.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-2,0,0,0,0,0,2,0,0,0,0,0,0,0,-4,0,0,0,0,0,16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(23)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(32.3873230598\)
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: \( 2 \)
Twist minimal: no (minimal twist has level 24)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 337.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 4056.337
Dual form 4056.2.c.e.337.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} +2.00000i q^{5} +1.00000 q^{9} +4.00000i q^{11} -2.00000i q^{15} -2.00000 q^{17} +4.00000i q^{19} +8.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} +6.00000 q^{29} -8.00000i q^{31} -4.00000i q^{33} +6.00000i q^{37} +6.00000i q^{41} -4.00000 q^{43} +2.00000i q^{45} +7.00000 q^{49} +2.00000 q^{51} -2.00000 q^{53} -8.00000 q^{55} -4.00000i q^{57} +4.00000i q^{59} -2.00000 q^{61} +4.00000i q^{67} -8.00000 q^{69} -8.00000i q^{71} +10.0000i q^{73} -1.00000 q^{75} -8.00000 q^{79} +1.00000 q^{81} +4.00000i q^{83} -4.00000i q^{85} -6.00000 q^{87} -6.00000i q^{89} +8.00000i q^{93} -8.00000 q^{95} -2.00000i q^{97} +4.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 2 q^{9} - 4 q^{17} + 16 q^{23} + 2 q^{25} - 2 q^{27} + 12 q^{29} - 8 q^{43} + 14 q^{49} + 4 q^{51} - 4 q^{53} - 16 q^{55} - 4 q^{61} - 16 q^{69} - 2 q^{75} - 16 q^{79} + 2 q^{81} - 12 q^{87}+ \cdots - 16 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4056\mathbb{Z}\right)^\times\).

\(n\) \(1015\) \(2029\) \(2705\) \(3889\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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
\(5\) 2.00000i 0.894427i 0.894427 + 0.447214i \(0.147584\pi\)
−0.894427 + 0.447214i \(0.852416\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 4.00000i 1.20605i 0.797724 + 0.603023i \(0.206037\pi\)
−0.797724 + 0.603023i \(0.793963\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 0 0
\(15\) − 2.00000i − 0.516398i
\(16\) 0 0
\(17\) −2.00000 −0.485071 −0.242536 − 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) 4.00000i 0.917663i 0.888523 + 0.458831i \(0.151732\pi\)
−0.888523 + 0.458831i \(0.848268\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 8.00000 1.66812 0.834058 − 0.551677i \(-0.186012\pi\)
0.834058 + 0.551677i \(0.186012\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 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
\(31\) − 8.00000i − 1.43684i −0.695608 − 0.718421i \(-0.744865\pi\)
0.695608 − 0.718421i \(-0.255135\pi\)
\(32\) 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
\(39\) 0 0
\(40\) 0 0
\(41\) 6.00000i 0.937043i 0.883452 + 0.468521i \(0.155213\pi\)
−0.883452 + 0.468521i \(0.844787\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 − 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) 2.00000i 0.298142i
\(46\) 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
\(53\) −2.00000 −0.274721 −0.137361 − 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 0 0
\(55\) −8.00000 −1.07872
\(56\) 0 0
\(57\) − 4.00000i − 0.529813i
\(58\) 0 0
\(59\) 4.00000i 0.520756i 0.965507 + 0.260378i \(0.0838471\pi\)
−0.965507 + 0.260378i \(0.916153\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 − 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 4.00000i 0.488678i 0.969690 + 0.244339i \(0.0785709\pi\)
−0.969690 + 0.244339i \(0.921429\pi\)
\(68\) 0 0
\(69\) −8.00000 −0.963087
\(70\) 0 0
\(71\) − 8.00000i − 0.949425i −0.880141 − 0.474713i \(-0.842552\pi\)
0.880141 − 0.474713i \(-0.157448\pi\)
\(72\) 0 0
\(73\) 10.0000i 1.17041i 0.810885 + 0.585206i \(0.198986\pi\)
−0.810885 + 0.585206i \(0.801014\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 − 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 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
\(85\) − 4.00000i − 0.433861i
\(86\) 0 0
\(87\) −6.00000 −0.643268
\(88\) 0 0
\(89\) − 6.00000i − 0.635999i −0.948091 − 0.317999i \(-0.896989\pi\)
0.948091 − 0.317999i \(-0.103011\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 8.00000i 0.829561i
\(94\) 0 0
\(95\) −8.00000 −0.820783
\(96\) 0 0
\(97\) − 2.00000i − 0.203069i −0.994832 − 0.101535i \(-0.967625\pi\)
0.994832 − 0.101535i \(-0.0323753\pi\)
\(98\) 0 0
\(99\) 4.00000i 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 4056.2.c.e.337.2 2
13.5 odd 4 4056.2.a.i.1.1 1
13.8 odd 4 24.2.a.a.1.1 ✓ 1
13.12 even 2 inner 4056.2.c.e.337.1 2
39.8 even 4 72.2.a.a.1.1 1
52.31 even 4 8112.2.a.be.1.1 1
52.47 even 4 48.2.a.a.1.1 1
65.8 even 4 600.2.f.e.49.1 2
65.34 odd 4 600.2.a.h.1.1 1
65.47 even 4 600.2.f.e.49.2 2
91.34 even 4 1176.2.a.i.1.1 1
91.47 even 12 1176.2.q.a.361.1 2
91.60 odd 12 1176.2.q.i.961.1 2
91.73 even 12 1176.2.q.a.961.1 2
91.86 odd 12 1176.2.q.i.361.1 2
104.21 odd 4 192.2.a.d.1.1 1
104.99 even 4 192.2.a.b.1.1 1
117.34 odd 12 648.2.i.g.433.1 2
117.47 even 12 648.2.i.b.433.1 2
117.86 even 12 648.2.i.b.217.1 2
117.112 odd 12 648.2.i.g.217.1 2
143.21 even 4 2904.2.a.c.1.1 1
156.47 odd 4 144.2.a.b.1.1 1
195.8 odd 4 1800.2.f.c.649.1 2
195.47 odd 4 1800.2.f.c.649.2 2
195.164 even 4 1800.2.a.m.1.1 1
208.21 odd 4 768.2.d.e.385.2 2
208.99 even 4 768.2.d.d.385.2 2
208.125 odd 4 768.2.d.e.385.1 2
208.203 even 4 768.2.d.d.385.1 2
221.203 odd 4 6936.2.a.p.1.1 1
247.151 even 4 8664.2.a.j.1.1 1
260.47 odd 4 1200.2.f.b.49.1 2
260.99 even 4 1200.2.a.d.1.1 1
260.203 odd 4 1200.2.f.b.49.2 2
273.47 odd 12 3528.2.s.y.361.1 2
273.86 even 12 3528.2.s.j.361.1 2
273.125 odd 4 3528.2.a.d.1.1 1
273.164 odd 12 3528.2.s.y.3313.1 2
273.242 even 12 3528.2.s.j.3313.1 2
312.125 even 4 576.2.a.d.1.1 1
312.203 odd 4 576.2.a.b.1.1 1
364.47 odd 12 2352.2.q.r.1537.1 2
364.151 even 12 2352.2.q.l.961.1 2
364.255 odd 12 2352.2.q.r.961.1 2
364.307 odd 4 2352.2.a.i.1.1 1
364.359 even 12 2352.2.q.l.1537.1 2
429.164 odd 4 8712.2.a.u.1.1 1
468.47 odd 12 1296.2.i.e.433.1 2
468.151 even 12 1296.2.i.m.433.1 2
468.203 odd 12 1296.2.i.e.865.1 2
468.463 even 12 1296.2.i.m.865.1 2
520.99 even 4 4800.2.a.cc.1.1 1
520.203 odd 4 4800.2.f.bg.3649.1 2
520.229 odd 4 4800.2.a.q.1.1 1
520.307 odd 4 4800.2.f.bg.3649.2 2
520.333 even 4 4800.2.f.d.3649.2 2
520.437 even 4 4800.2.f.d.3649.1 2
572.307 odd 4 5808.2.a.s.1.1 1
624.125 even 4 2304.2.d.i.1153.1 2
624.203 odd 4 2304.2.d.k.1153.2 2
624.437 even 4 2304.2.d.i.1153.2 2
624.515 odd 4 2304.2.d.k.1153.1 2
728.125 even 4 9408.2.a.h.1.1 1
728.307 odd 4 9408.2.a.cc.1.1 1
780.47 even 4 3600.2.f.r.2449.2 2
780.203 even 4 3600.2.f.r.2449.1 2
780.359 odd 4 3600.2.a.v.1.1 1
1092.671 even 4 7056.2.a.q.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
24.2.a.a.1.1 ✓ 1 13.8 odd 4
48.2.a.a.1.1 1 52.47 even 4
72.2.a.a.1.1 1 39.8 even 4
144.2.a.b.1.1 1 156.47 odd 4
192.2.a.b.1.1 1 104.99 even 4
192.2.a.d.1.1 1 104.21 odd 4
576.2.a.b.1.1 1 312.203 odd 4
576.2.a.d.1.1 1 312.125 even 4
600.2.a.h.1.1 1 65.34 odd 4
600.2.f.e.49.1 2 65.8 even 4
600.2.f.e.49.2 2 65.47 even 4
648.2.i.b.217.1 2 117.86 even 12
648.2.i.b.433.1 2 117.47 even 12
648.2.i.g.217.1 2 117.112 odd 12
648.2.i.g.433.1 2 117.34 odd 12
768.2.d.d.385.1 2 208.203 even 4
768.2.d.d.385.2 2 208.99 even 4
768.2.d.e.385.1 2 208.125 odd 4
768.2.d.e.385.2 2 208.21 odd 4
1176.2.a.i.1.1 1 91.34 even 4
1176.2.q.a.361.1 2 91.47 even 12
1176.2.q.a.961.1 2 91.73 even 12
1176.2.q.i.361.1 2 91.86 odd 12
1176.2.q.i.961.1 2 91.60 odd 12
1200.2.a.d.1.1 1 260.99 even 4
1200.2.f.b.49.1 2 260.47 odd 4
1200.2.f.b.49.2 2 260.203 odd 4
1296.2.i.e.433.1 2 468.47 odd 12
1296.2.i.e.865.1 2 468.203 odd 12
1296.2.i.m.433.1 2 468.151 even 12
1296.2.i.m.865.1 2 468.463 even 12
1800.2.a.m.1.1 1 195.164 even 4
1800.2.f.c.649.1 2 195.8 odd 4
1800.2.f.c.649.2 2 195.47 odd 4
2304.2.d.i.1153.1 2 624.125 even 4
2304.2.d.i.1153.2 2 624.437 even 4
2304.2.d.k.1153.1 2 624.515 odd 4
2304.2.d.k.1153.2 2 624.203 odd 4
2352.2.a.i.1.1 1 364.307 odd 4
2352.2.q.l.961.1 2 364.151 even 12
2352.2.q.l.1537.1 2 364.359 even 12
2352.2.q.r.961.1 2 364.255 odd 12
2352.2.q.r.1537.1 2 364.47 odd 12
2904.2.a.c.1.1 1 143.21 even 4
3528.2.a.d.1.1 1 273.125 odd 4
3528.2.s.j.361.1 2 273.86 even 12
3528.2.s.j.3313.1 2 273.242 even 12
3528.2.s.y.361.1 2 273.47 odd 12
3528.2.s.y.3313.1 2 273.164 odd 12
3600.2.a.v.1.1 1 780.359 odd 4
3600.2.f.r.2449.1 2 780.203 even 4
3600.2.f.r.2449.2 2 780.47 even 4
4056.2.a.i.1.1 1 13.5 odd 4
4056.2.c.e.337.1 2 13.12 even 2 inner
4056.2.c.e.337.2 2 1.1 even 1 trivial
4800.2.a.q.1.1 1 520.229 odd 4
4800.2.a.cc.1.1 1 520.99 even 4
4800.2.f.d.3649.1 2 520.437 even 4
4800.2.f.d.3649.2 2 520.333 even 4
4800.2.f.bg.3649.1 2 520.203 odd 4
4800.2.f.bg.3649.2 2 520.307 odd 4
5808.2.a.s.1.1 1 572.307 odd 4
6936.2.a.p.1.1 1 221.203 odd 4
7056.2.a.q.1.1 1 1092.671 even 4
8112.2.a.be.1.1 1 52.31 even 4
8664.2.a.j.1.1 1 247.151 even 4
8712.2.a.u.1.1 1 429.164 odd 4
9408.2.a.h.1.1 1 728.125 even 4
9408.2.a.cc.1.1 1 728.307 odd 4