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Properties

Label 60.2.d.a.49.1
Level $60$
Weight $2$
Character 60.49
Analytic conductor $0.479$
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] = [60,2,Mod(49,60)] 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("60.49"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(60, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 60 = 2^{2} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 60.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.479102412128\)
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: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 60.49
Dual form 60.2.d.a.49.2

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000i q^{3} +(1.00000 - 2.00000i) q^{5} +4.00000i q^{7} -1.00000 q^{9} -4.00000 q^{11} +(-2.00000 - 1.00000i) q^{15} +4.00000i q^{17} +4.00000 q^{21} -4.00000i q^{23} +(-3.00000 - 4.00000i) q^{25} +1.00000i q^{27} +6.00000 q^{29} +4.00000 q^{31} +4.00000i q^{33} +(8.00000 + 4.00000i) q^{35} -8.00000i q^{37} -10.0000 q^{41} -4.00000i q^{43} +(-1.00000 + 2.00000i) q^{45} -4.00000i q^{47} -9.00000 q^{49} +4.00000 q^{51} +12.0000i q^{53} +(-4.00000 + 8.00000i) q^{55} -4.00000 q^{59} +2.00000 q^{61} -4.00000i q^{63} -4.00000i q^{67} -4.00000 q^{69} +8.00000i q^{73} +(-4.00000 + 3.00000i) q^{75} -16.0000i q^{77} +12.0000 q^{79} +1.00000 q^{81} -4.00000i q^{83} +(8.00000 + 4.00000i) q^{85} -6.00000i q^{87} +10.0000 q^{89} -4.00000i q^{93} +8.00000i q^{97} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} - 2 q^{9} - 8 q^{11} - 4 q^{15} + 8 q^{21} - 6 q^{25} + 12 q^{29} + 8 q^{31} + 16 q^{35} - 20 q^{41} - 2 q^{45} - 18 q^{49} + 8 q^{51} - 8 q^{55} - 8 q^{59} + 4 q^{61} - 8 q^{69} - 8 q^{75}+ \cdots + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/60\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(37\) \(41\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(5\) 1.00000 − 2.00000i 0.447214 − 0.894427i
\(6\) 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
\(11\) −4.00000 −1.20605 −0.603023 − 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) −2.00000 − 1.00000i −0.516398 − 0.258199i
\(16\) 0 0
\(17\) 4.00000i 0.970143i 0.874475 + 0.485071i \(0.161206\pi\)
−0.874475 + 0.485071i \(0.838794\pi\)
\(18\) 0 0
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 4.00000 0.872872
\(22\) 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
\(27\) 1.00000i 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
\(31\) 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
\(35\) 8.00000 + 4.00000i 1.35225 + 0.676123i
\(36\) 0 0
\(37\) − 8.00000i − 1.31519i −0.753371 − 0.657596i \(-0.771573\pi\)
0.753371 − 0.657596i \(-0.228427\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −10.0000 −1.56174 −0.780869 − 0.624695i \(-0.785223\pi\)
−0.780869 + 0.624695i \(0.785223\pi\)
\(42\) 0 0
\(43\) − 4.00000i − 0.609994i −0.952353 − 0.304997i \(-0.901344\pi\)
0.952353 − 0.304997i \(-0.0986555\pi\)
\(44\) 0 0
\(45\) −1.00000 + 2.00000i −0.149071 + 0.298142i
\(46\) 0 0
\(47\) − 4.00000i − 0.583460i −0.956501 − 0.291730i \(-0.905769\pi\)
0.956501 − 0.291730i \(-0.0942309\pi\)
\(48\) 0 0
\(49\) −9.00000 −1.28571
\(50\) 0 0
\(51\) 4.00000 0.560112
\(52\) 0 0
\(53\) 12.0000i 1.64833i 0.566352 + 0.824163i \(0.308354\pi\)
−0.566352 + 0.824163i \(0.691646\pi\)
\(54\) 0 0
\(55\) −4.00000 + 8.00000i −0.539360 + 1.07872i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −4.00000 −0.520756 −0.260378 − 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 − 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 0 0
\(63\) − 4.00000i − 0.503953i
\(64\) 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
\(69\) −4.00000 −0.481543
\(70\) 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
\(75\) −4.00000 + 3.00000i −0.461880 + 0.346410i
\(76\) 0 0
\(77\) − 16.0000i − 1.82337i
\(78\) 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
\(83\) − 4.00000i − 0.439057i −0.975606 − 0.219529i \(-0.929548\pi\)
0.975606 − 0.219529i \(-0.0704519\pi\)
\(84\) 0 0
\(85\) 8.00000 + 4.00000i 0.867722 + 0.433861i
\(86\) 0 0
\(87\) − 6.00000i − 0.643268i
\(88\) 0 0
\(89\) 10.0000 1.06000 0.529999 − 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) − 4.00000i − 0.414781i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 8.00000i 0.812277i 0.913812 + 0.406138i \(0.133125\pi\)
−0.913812 + 0.406138i \(0.866875\pi\)
\(98\) 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 60.2.d.a.49.1 ✓ 2
3.2 odd 2 180.2.d.a.109.2 2
4.3 odd 2 240.2.f.b.49.2 2
5.2 odd 4 300.2.a.a.1.1 1
5.3 odd 4 300.2.a.d.1.1 1
5.4 even 2 inner 60.2.d.a.49.2 yes 2
7.2 even 3 2940.2.bb.d.949.2 4
7.3 odd 6 2940.2.bb.e.1549.2 4
7.4 even 3 2940.2.bb.d.1549.1 4
7.5 odd 6 2940.2.bb.e.949.1 4
7.6 odd 2 2940.2.k.c.589.2 2
8.3 odd 2 960.2.f.c.769.1 2
8.5 even 2 960.2.f.f.769.2 2
9.2 odd 6 1620.2.r.d.109.1 4
9.4 even 3 1620.2.r.c.1189.1 4
9.5 odd 6 1620.2.r.d.1189.2 4
9.7 even 3 1620.2.r.c.109.2 4
12.11 even 2 720.2.f.c.289.2 2
15.2 even 4 900.2.a.a.1.1 1
15.8 even 4 900.2.a.h.1.1 1
15.14 odd 2 180.2.d.a.109.1 2
16.3 odd 4 3840.2.d.b.2689.1 2
16.5 even 4 3840.2.d.o.2689.2 2
16.11 odd 4 3840.2.d.be.2689.2 2
16.13 even 4 3840.2.d.r.2689.1 2
20.3 even 4 1200.2.a.a.1.1 1
20.7 even 4 1200.2.a.s.1.1 1
20.19 odd 2 240.2.f.b.49.1 2
24.5 odd 2 2880.2.f.l.1729.1 2
24.11 even 2 2880.2.f.p.1729.1 2
35.4 even 6 2940.2.bb.d.1549.2 4
35.9 even 6 2940.2.bb.d.949.1 4
35.19 odd 6 2940.2.bb.e.949.2 4
35.24 odd 6 2940.2.bb.e.1549.1 4
35.34 odd 2 2940.2.k.c.589.1 2
40.3 even 4 4800.2.a.bk.1.1 1
40.13 odd 4 4800.2.a.bj.1.1 1
40.19 odd 2 960.2.f.c.769.2 2
40.27 even 4 4800.2.a.bf.1.1 1
40.29 even 2 960.2.f.f.769.1 2
40.37 odd 4 4800.2.a.bn.1.1 1
45.4 even 6 1620.2.r.c.1189.2 4
45.14 odd 6 1620.2.r.d.1189.1 4
45.29 odd 6 1620.2.r.d.109.2 4
45.34 even 6 1620.2.r.c.109.1 4
60.23 odd 4 3600.2.a.d.1.1 1
60.47 odd 4 3600.2.a.bm.1.1 1
60.59 even 2 720.2.f.c.289.1 2
80.19 odd 4 3840.2.d.be.2689.1 2
80.29 even 4 3840.2.d.o.2689.1 2
80.59 odd 4 3840.2.d.b.2689.2 2
80.69 even 4 3840.2.d.r.2689.2 2
120.29 odd 2 2880.2.f.l.1729.2 2
120.59 even 2 2880.2.f.p.1729.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
60.2.d.a.49.1 ✓ 2 1.1 even 1 trivial
60.2.d.a.49.2 yes 2 5.4 even 2 inner
180.2.d.a.109.1 2 15.14 odd 2
180.2.d.a.109.2 2 3.2 odd 2
240.2.f.b.49.1 2 20.19 odd 2
240.2.f.b.49.2 2 4.3 odd 2
300.2.a.a.1.1 1 5.2 odd 4
300.2.a.d.1.1 1 5.3 odd 4
720.2.f.c.289.1 2 60.59 even 2
720.2.f.c.289.2 2 12.11 even 2
900.2.a.a.1.1 1 15.2 even 4
900.2.a.h.1.1 1 15.8 even 4
960.2.f.c.769.1 2 8.3 odd 2
960.2.f.c.769.2 2 40.19 odd 2
960.2.f.f.769.1 2 40.29 even 2
960.2.f.f.769.2 2 8.5 even 2
1200.2.a.a.1.1 1 20.3 even 4
1200.2.a.s.1.1 1 20.7 even 4
1620.2.r.c.109.1 4 45.34 even 6
1620.2.r.c.109.2 4 9.7 even 3
1620.2.r.c.1189.1 4 9.4 even 3
1620.2.r.c.1189.2 4 45.4 even 6
1620.2.r.d.109.1 4 9.2 odd 6
1620.2.r.d.109.2 4 45.29 odd 6
1620.2.r.d.1189.1 4 45.14 odd 6
1620.2.r.d.1189.2 4 9.5 odd 6
2880.2.f.l.1729.1 2 24.5 odd 2
2880.2.f.l.1729.2 2 120.29 odd 2
2880.2.f.p.1729.1 2 24.11 even 2
2880.2.f.p.1729.2 2 120.59 even 2
2940.2.k.c.589.1 2 35.34 odd 2
2940.2.k.c.589.2 2 7.6 odd 2
2940.2.bb.d.949.1 4 35.9 even 6
2940.2.bb.d.949.2 4 7.2 even 3
2940.2.bb.d.1549.1 4 7.4 even 3
2940.2.bb.d.1549.2 4 35.4 even 6
2940.2.bb.e.949.1 4 7.5 odd 6
2940.2.bb.e.949.2 4 35.19 odd 6
2940.2.bb.e.1549.1 4 35.24 odd 6
2940.2.bb.e.1549.2 4 7.3 odd 6
3600.2.a.d.1.1 1 60.23 odd 4
3600.2.a.bm.1.1 1 60.47 odd 4
3840.2.d.b.2689.1 2 16.3 odd 4
3840.2.d.b.2689.2 2 80.59 odd 4
3840.2.d.o.2689.1 2 80.29 even 4
3840.2.d.o.2689.2 2 16.5 even 4
3840.2.d.r.2689.1 2 16.13 even 4
3840.2.d.r.2689.2 2 80.69 even 4
3840.2.d.be.2689.1 2 80.19 odd 4
3840.2.d.be.2689.2 2 16.11 odd 4
4800.2.a.bf.1.1 1 40.27 even 4
4800.2.a.bj.1.1 1 40.13 odd 4
4800.2.a.bk.1.1 1 40.3 even 4
4800.2.a.bn.1.1 1 40.37 odd 4