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Properties

Label 60.2.h.a.59.4
Level $60$
Weight $2$
Character 60.59
Analytic conductor $0.479$
Analytic rank $0$
Dimension $4$
CM discriminant -20
Inner twists $8$

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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(59,60)] 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("60.59"); 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([1, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 60 = 2^{2} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 60.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.479102412128\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-5})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 4x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 59.4
Root \(-1.58114 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 60.59
Dual form 60.2.h.a.59.2

$q$-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)
 
\(f(q)\) \(=\) \(q+1.41421i q^{2} +(1.58114 + 0.707107i) q^{3} -2.00000 q^{4} -2.23607i q^{5} +(-1.00000 + 2.23607i) q^{6} -3.16228 q^{7} -2.82843i q^{8} +(2.00000 + 2.23607i) q^{9} +3.16228 q^{10} +(-3.16228 - 1.41421i) q^{12} -4.47214i q^{14} +(1.58114 - 3.53553i) q^{15} +4.00000 q^{16} +(-3.16228 + 2.82843i) q^{18} +4.47214i q^{20} +(-5.00000 - 2.23607i) q^{21} -1.41421i q^{23} +(2.00000 - 4.47214i) q^{24} -5.00000 q^{25} +(1.58114 + 4.94975i) q^{27} +6.32456 q^{28} +8.94427i q^{29} +(5.00000 + 2.23607i) q^{30} +5.65685i q^{32} +7.07107i q^{35} +(-4.00000 - 4.47214i) q^{36} -6.32456 q^{40} -4.47214i q^{41} +(3.16228 - 7.07107i) q^{42} -3.16228 q^{43} +(5.00000 - 4.47214i) q^{45} +2.00000 q^{46} -9.89949i q^{47} +(6.32456 + 2.82843i) q^{48} +3.00000 q^{49} -7.07107i q^{50} +(-7.00000 + 2.23607i) q^{54} +8.94427i q^{56} -12.6491 q^{58} +(-3.16228 + 7.07107i) q^{60} +8.00000 q^{61} +(-6.32456 - 7.07107i) q^{63} -8.00000 q^{64} +15.8114 q^{67} +(1.00000 - 2.23607i) q^{69} -10.0000 q^{70} +(6.32456 - 5.65685i) q^{72} +(-7.90569 - 3.53553i) q^{75} -8.94427i q^{80} +(-1.00000 + 8.94427i) q^{81} +6.32456 q^{82} +15.5563i q^{83} +(10.0000 + 4.47214i) q^{84} -4.47214i q^{86} +(-6.32456 + 14.1421i) q^{87} -17.8885i q^{89} +(6.32456 + 7.07107i) q^{90} +2.82843i q^{92} +14.0000 q^{94} +(-4.00000 + 8.94427i) q^{96} +4.24264i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{4} - 4 q^{6} + 8 q^{9} + 16 q^{16} - 20 q^{21} + 8 q^{24} - 20 q^{25} + 20 q^{30} - 16 q^{36} + 20 q^{45} + 8 q^{46} + 12 q^{49} - 28 q^{54} + 32 q^{61} - 32 q^{64} + 4 q^{69} - 40 q^{70} - 4 q^{81}+ \cdots - 16 q^{96}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.41421i 1.00000i
\(3\) 1.58114 + 0.707107i 0.912871 + 0.408248i
\(4\) −2.00000 −1.00000
\(5\) − 2.23607i − 1.00000i
\(6\) −1.00000 + 2.23607i −0.408248 + 0.912871i
\(7\) −3.16228 −1.19523 −0.597614 − 0.801784i \(-0.703885\pi\)
−0.597614 + 0.801784i \(0.703885\pi\)
\(8\) − 2.82843i − 1.00000i
\(9\) 2.00000 + 2.23607i 0.666667 + 0.745356i
\(10\) 3.16228 1.00000
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) −3.16228 − 1.41421i −0.912871 − 0.408248i
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) − 4.47214i − 1.19523i
\(15\) 1.58114 − 3.53553i 0.408248 − 0.912871i
\(16\) 4.00000 1.00000
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) −3.16228 + 2.82843i −0.745356 + 0.666667i
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 4.47214i 1.00000i
\(21\) −5.00000 − 2.23607i −1.09109 − 0.487950i
\(22\) 0 0
\(23\) − 1.41421i − 0.294884i −0.989071 − 0.147442i \(-0.952896\pi\)
0.989071 − 0.147442i \(-0.0471040\pi\)
\(24\) 2.00000 − 4.47214i 0.408248 − 0.912871i
\(25\) −5.00000 −1.00000
\(26\) 0 0
\(27\) 1.58114 + 4.94975i 0.304290 + 0.952579i
\(28\) 6.32456 1.19523
\(29\) 8.94427i 1.66091i 0.557086 + 0.830455i \(0.311919\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 5.00000 + 2.23607i 0.912871 + 0.408248i
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 5.65685i 1.00000i
\(33\) 0 0
\(34\) 0 0
\(35\) 7.07107i 1.19523i
\(36\) −4.00000 − 4.47214i −0.666667 − 0.745356i
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) −6.32456 −1.00000
\(41\) − 4.47214i − 0.698430i −0.937043 − 0.349215i \(-0.886448\pi\)
0.937043 − 0.349215i \(-0.113552\pi\)
\(42\) 3.16228 − 7.07107i 0.487950 − 1.09109i
\(43\) −3.16228 −0.482243 −0.241121 − 0.970495i \(-0.577515\pi\)
−0.241121 + 0.970495i \(0.577515\pi\)
\(44\) 0 0
\(45\) 5.00000 − 4.47214i 0.745356 − 0.666667i
\(46\) 2.00000 0.294884
\(47\) − 9.89949i − 1.44399i −0.691898 − 0.721995i \(-0.743225\pi\)
0.691898 − 0.721995i \(-0.256775\pi\)
\(48\) 6.32456 + 2.82843i 0.912871 + 0.408248i
\(49\) 3.00000 0.428571
\(50\) − 7.07107i − 1.00000i
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) −7.00000 + 2.23607i −0.952579 + 0.304290i
\(55\) 0 0
\(56\) 8.94427i 1.19523i
\(57\) 0 0
\(58\) −12.6491 −1.66091
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) −3.16228 + 7.07107i −0.408248 + 0.912871i
\(61\) 8.00000 1.02430 0.512148 − 0.858898i \(-0.328850\pi\)
0.512148 + 0.858898i \(0.328850\pi\)
\(62\) 0 0
\(63\) −6.32456 − 7.07107i −0.796819 − 0.890871i
\(64\) −8.00000 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 15.8114 1.93167 0.965834 − 0.259161i \(-0.0834459\pi\)
0.965834 + 0.259161i \(0.0834459\pi\)
\(68\) 0 0
\(69\) 1.00000 − 2.23607i 0.120386 − 0.269191i
\(70\) −10.0000 −1.19523
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 6.32456 − 5.65685i 0.745356 − 0.666667i
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) −7.90569 − 3.53553i −0.912871 − 0.408248i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) − 8.94427i − 1.00000i
\(81\) −1.00000 + 8.94427i −0.111111 + 0.993808i
\(82\) 6.32456 0.698430
\(83\) 15.5563i 1.70753i 0.520658 + 0.853766i \(0.325687\pi\)
−0.520658 + 0.853766i \(0.674313\pi\)
\(84\) 10.0000 + 4.47214i 1.09109 + 0.487950i
\(85\) 0 0
\(86\) − 4.47214i − 0.482243i
\(87\) −6.32456 + 14.1421i −0.678064 + 1.51620i
\(88\) 0 0
\(89\) − 17.8885i − 1.89618i −0.317999 − 0.948091i \(-0.603011\pi\)
0.317999 − 0.948091i \(-0.396989\pi\)
\(90\) 6.32456 + 7.07107i 0.666667 + 0.745356i
\(91\) 0 0
\(92\) 2.82843i 0.294884i
\(93\) 0 0
\(94\) 14.0000 1.44399
\(95\) 0 0
\(96\) −4.00000 + 8.94427i −0.408248 + 0.912871i
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 4.24264i 0.428571i
\(99\) 0 0
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.h.a.59.4 yes 4
3.2 odd 2 inner 60.2.h.a.59.2 yes 4
4.3 odd 2 inner 60.2.h.a.59.1 ✓ 4
5.2 odd 4 300.2.e.b.251.1 4
5.3 odd 4 300.2.e.b.251.4 4
5.4 even 2 inner 60.2.h.a.59.1 ✓ 4
8.3 odd 2 960.2.o.c.959.4 4
8.5 even 2 960.2.o.c.959.1 4
12.11 even 2 inner 60.2.h.a.59.3 yes 4
15.2 even 4 300.2.e.b.251.3 4
15.8 even 4 300.2.e.b.251.2 4
15.14 odd 2 inner 60.2.h.a.59.3 yes 4
20.3 even 4 300.2.e.b.251.1 4
20.7 even 4 300.2.e.b.251.4 4
20.19 odd 2 CM 60.2.h.a.59.4 yes 4
24.5 odd 2 960.2.o.c.959.2 4
24.11 even 2 960.2.o.c.959.3 4
40.19 odd 2 960.2.o.c.959.1 4
40.29 even 2 960.2.o.c.959.4 4
60.23 odd 4 300.2.e.b.251.3 4
60.47 odd 4 300.2.e.b.251.2 4
60.59 even 2 inner 60.2.h.a.59.2 yes 4
120.29 odd 2 960.2.o.c.959.3 4
120.59 even 2 960.2.o.c.959.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
60.2.h.a.59.1 ✓ 4 4.3 odd 2 inner
60.2.h.a.59.1 ✓ 4 5.4 even 2 inner
60.2.h.a.59.2 yes 4 3.2 odd 2 inner
60.2.h.a.59.2 yes 4 60.59 even 2 inner
60.2.h.a.59.3 yes 4 12.11 even 2 inner
60.2.h.a.59.3 yes 4 15.14 odd 2 inner
60.2.h.a.59.4 yes 4 1.1 even 1 trivial
60.2.h.a.59.4 yes 4 20.19 odd 2 CM
300.2.e.b.251.1 4 5.2 odd 4
300.2.e.b.251.1 4 20.3 even 4
300.2.e.b.251.2 4 15.8 even 4
300.2.e.b.251.2 4 60.47 odd 4
300.2.e.b.251.3 4 15.2 even 4
300.2.e.b.251.3 4 60.23 odd 4
300.2.e.b.251.4 4 5.3 odd 4
300.2.e.b.251.4 4 20.7 even 4
960.2.o.c.959.1 4 8.5 even 2
960.2.o.c.959.1 4 40.19 odd 2
960.2.o.c.959.2 4 24.5 odd 2
960.2.o.c.959.2 4 120.59 even 2
960.2.o.c.959.3 4 24.11 even 2
960.2.o.c.959.3 4 120.29 odd 2
960.2.o.c.959.4 4 8.3 odd 2
960.2.o.c.959.4 4 40.29 even 2