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Properties

Label 60.3.b.a.29.1
Level $60$
Weight $3$
Character 60.29
Analytic conductor $1.635$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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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] = [60,3,Mod(29,60)] 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("60.29"); S:= CuspForms(chi, 3); 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, 1, 1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 60 = 2^{2} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 60.b (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: \(1.63488158616\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 29.1
Root \(-1.61803i\) of defining polynomial
Character \(\chi\) \(=\) 60.29
Dual form 60.3.b.a.29.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(-2.23607 - 2.00000i) q^{3} +(-2.23607 - 4.47214i) q^{5} -8.00000i q^{7} +(1.00000 + 8.94427i) q^{9} -8.94427i q^{11} +12.0000i q^{13} +(-3.94427 + 14.4721i) q^{15} +31.3050 q^{17} -6.00000 q^{19} +(-16.0000 + 17.8885i) q^{21} -4.47214 q^{23} +(-15.0000 + 20.0000i) q^{25} +(15.6525 - 22.0000i) q^{27} -26.8328i q^{29} +34.0000 q^{31} +(-17.8885 + 20.0000i) q^{33} +(-35.7771 + 17.8885i) q^{35} -44.0000i q^{37} +(24.0000 - 26.8328i) q^{39} +17.8885i q^{41} -28.0000i q^{43} +(37.7639 - 24.4721i) q^{45} -4.47214 q^{47} -15.0000 q^{49} +(-70.0000 - 62.6099i) q^{51} -40.2492 q^{53} +(-40.0000 + 20.0000i) q^{55} +(13.4164 + 12.0000i) q^{57} +98.3870i q^{59} +74.0000 q^{61} +(71.5542 - 8.00000i) q^{63} +(53.6656 - 26.8328i) q^{65} +92.0000i q^{67} +(10.0000 + 8.94427i) q^{69} +53.6656i q^{71} +56.0000i q^{73} +(73.5410 - 14.7214i) q^{75} -71.5542 q^{77} -78.0000 q^{79} +(-79.0000 + 17.8885i) q^{81} +102.859 q^{83} +(-70.0000 - 140.000i) q^{85} +(-53.6656 + 60.0000i) q^{87} -17.8885i q^{89} +96.0000 q^{91} +(-76.0263 - 68.0000i) q^{93} +(13.4164 + 26.8328i) q^{95} +32.0000i q^{97} +(80.0000 - 8.94427i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{9} + 20 q^{15} - 24 q^{19} - 64 q^{21} - 60 q^{25} + 136 q^{31} + 96 q^{39} + 160 q^{45} - 60 q^{49} - 280 q^{51} - 160 q^{55} + 296 q^{61} + 40 q^{69} + 160 q^{75} - 312 q^{79} - 316 q^{81}+ \cdots + 320 q^{99}+O(q^{100}) \) 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
\(3\) −2.23607 − 2.00000i −0.745356 − 0.666667i
\(4\) 0 0
\(5\) −2.23607 − 4.47214i −0.447214 − 0.894427i
\(6\) 0 0
\(7\) − 8.00000i − 1.14286i −0.820652 − 0.571429i \(-0.806389\pi\)
0.820652 − 0.571429i \(-0.193611\pi\)
\(8\) 0 0
\(9\) 1.00000 + 8.94427i 0.111111 + 0.993808i
\(10\) 0 0
\(11\) − 8.94427i − 0.813116i −0.913625 − 0.406558i \(-0.866729\pi\)
0.913625 − 0.406558i \(-0.133271\pi\)
\(12\) 0 0
\(13\) 12.0000i 0.923077i 0.887120 + 0.461538i \(0.152702\pi\)
−0.887120 + 0.461538i \(0.847298\pi\)
\(14\) 0 0
\(15\) −3.94427 + 14.4721i −0.262951 + 0.964809i
\(16\) 0 0
\(17\) 31.3050 1.84147 0.920734 − 0.390191i \(-0.127591\pi\)
0.920734 + 0.390191i \(0.127591\pi\)
\(18\) 0 0
\(19\) −6.00000 −0.315789 −0.157895 − 0.987456i \(-0.550471\pi\)
−0.157895 + 0.987456i \(0.550471\pi\)
\(20\) 0 0
\(21\) −16.0000 + 17.8885i −0.761905 + 0.851835i
\(22\) 0 0
\(23\) −4.47214 −0.194441 −0.0972203 − 0.995263i \(-0.530995\pi\)
−0.0972203 + 0.995263i \(0.530995\pi\)
\(24\) 0 0
\(25\) −15.0000 + 20.0000i −0.600000 + 0.800000i
\(26\) 0 0
\(27\) 15.6525 − 22.0000i 0.579721 − 0.814815i
\(28\) 0 0
\(29\) − 26.8328i − 0.925270i −0.886549 − 0.462635i \(-0.846904\pi\)
0.886549 − 0.462635i \(-0.153096\pi\)
\(30\) 0 0
\(31\) 34.0000 1.09677 0.548387 − 0.836225i \(-0.315242\pi\)
0.548387 + 0.836225i \(0.315242\pi\)
\(32\) 0 0
\(33\) −17.8885 + 20.0000i −0.542077 + 0.606061i
\(34\) 0 0
\(35\) −35.7771 + 17.8885i −1.02220 + 0.511101i
\(36\) 0 0
\(37\) − 44.0000i − 1.18919i −0.804026 − 0.594595i \(-0.797313\pi\)
0.804026 − 0.594595i \(-0.202687\pi\)
\(38\) 0 0
\(39\) 24.0000 − 26.8328i 0.615385 − 0.688021i
\(40\) 0 0
\(41\) 17.8885i 0.436306i 0.975915 + 0.218153i \(0.0700032\pi\)
−0.975915 + 0.218153i \(0.929997\pi\)
\(42\) 0 0
\(43\) − 28.0000i − 0.651163i −0.945514 − 0.325581i \(-0.894440\pi\)
0.945514 − 0.325581i \(-0.105560\pi\)
\(44\) 0 0
\(45\) 37.7639 − 24.4721i 0.839198 − 0.543825i
\(46\) 0 0
\(47\) −4.47214 −0.0951518 −0.0475759 − 0.998868i \(-0.515150\pi\)
−0.0475759 + 0.998868i \(0.515150\pi\)
\(48\) 0 0
\(49\) −15.0000 −0.306122
\(50\) 0 0
\(51\) −70.0000 − 62.6099i −1.37255 − 1.22765i
\(52\) 0 0
\(53\) −40.2492 −0.759419 −0.379710 − 0.925106i \(-0.623976\pi\)
−0.379710 + 0.925106i \(0.623976\pi\)
\(54\) 0 0
\(55\) −40.0000 + 20.0000i −0.727273 + 0.363636i
\(56\) 0 0
\(57\) 13.4164 + 12.0000i 0.235376 + 0.210526i
\(58\) 0 0
\(59\) 98.3870i 1.66758i 0.552085 + 0.833788i \(0.313833\pi\)
−0.552085 + 0.833788i \(0.686167\pi\)
\(60\) 0 0
\(61\) 74.0000 1.21311 0.606557 − 0.795040i \(-0.292550\pi\)
0.606557 + 0.795040i \(0.292550\pi\)
\(62\) 0 0
\(63\) 71.5542 − 8.00000i 1.13578 − 0.126984i
\(64\) 0 0
\(65\) 53.6656 − 26.8328i 0.825625 − 0.412813i
\(66\) 0 0
\(67\) 92.0000i 1.37313i 0.727066 + 0.686567i \(0.240883\pi\)
−0.727066 + 0.686567i \(0.759117\pi\)
\(68\) 0 0
\(69\) 10.0000 + 8.94427i 0.144928 + 0.129627i
\(70\) 0 0
\(71\) 53.6656i 0.755854i 0.925835 + 0.377927i \(0.123363\pi\)
−0.925835 + 0.377927i \(0.876637\pi\)
\(72\) 0 0
\(73\) 56.0000i 0.767123i 0.923515 + 0.383562i \(0.125303\pi\)
−0.923515 + 0.383562i \(0.874697\pi\)
\(74\) 0 0
\(75\) 73.5410 − 14.7214i 0.980547 − 0.196285i
\(76\) 0 0
\(77\) −71.5542 −0.929275
\(78\) 0 0
\(79\) −78.0000 −0.987342 −0.493671 − 0.869649i \(-0.664345\pi\)
−0.493671 + 0.869649i \(0.664345\pi\)
\(80\) 0 0
\(81\) −79.0000 + 17.8885i −0.975309 + 0.220846i
\(82\) 0 0
\(83\) 102.859 1.23927 0.619633 − 0.784891i \(-0.287281\pi\)
0.619633 + 0.784891i \(0.287281\pi\)
\(84\) 0 0
\(85\) −70.0000 − 140.000i −0.823529 − 1.64706i
\(86\) 0 0
\(87\) −53.6656 + 60.0000i −0.616846 + 0.689655i
\(88\) 0 0
\(89\) − 17.8885i − 0.200995i −0.994937 − 0.100497i \(-0.967957\pi\)
0.994937 − 0.100497i \(-0.0320434\pi\)
\(90\) 0 0
\(91\) 96.0000 1.05495
\(92\) 0 0
\(93\) −76.0263 − 68.0000i −0.817487 − 0.731183i
\(94\) 0 0
\(95\) 13.4164 + 26.8328i 0.141225 + 0.282451i
\(96\) 0 0
\(97\) 32.0000i 0.329897i 0.986302 + 0.164948i \(0.0527458\pi\)
−0.986302 + 0.164948i \(0.947254\pi\)
\(98\) 0 0
\(99\) 80.0000 − 8.94427i 0.808081 − 0.0903462i
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 60.3.b.a.29.1 ✓ 4
3.2 odd 2 inner 60.3.b.a.29.3 yes 4
4.3 odd 2 240.3.c.d.209.4 4
5.2 odd 4 300.3.g.e.101.2 2
5.3 odd 4 300.3.g.h.101.1 2
5.4 even 2 inner 60.3.b.a.29.4 yes 4
8.3 odd 2 960.3.c.g.449.1 4
8.5 even 2 960.3.c.h.449.4 4
9.2 odd 6 1620.3.t.b.1349.1 8
9.4 even 3 1620.3.t.b.269.2 8
9.5 odd 6 1620.3.t.b.269.3 8
9.7 even 3 1620.3.t.b.1349.4 8
12.11 even 2 240.3.c.d.209.2 4
15.2 even 4 300.3.g.e.101.1 2
15.8 even 4 300.3.g.h.101.2 2
15.14 odd 2 inner 60.3.b.a.29.2 yes 4
20.3 even 4 1200.3.l.h.401.2 2
20.7 even 4 1200.3.l.q.401.1 2
20.19 odd 2 240.3.c.d.209.1 4
24.5 odd 2 960.3.c.h.449.2 4
24.11 even 2 960.3.c.g.449.3 4
40.19 odd 2 960.3.c.g.449.4 4
40.29 even 2 960.3.c.h.449.1 4
45.4 even 6 1620.3.t.b.269.1 8
45.14 odd 6 1620.3.t.b.269.4 8
45.29 odd 6 1620.3.t.b.1349.2 8
45.34 even 6 1620.3.t.b.1349.3 8
60.23 odd 4 1200.3.l.h.401.1 2
60.47 odd 4 1200.3.l.q.401.2 2
60.59 even 2 240.3.c.d.209.3 4
120.29 odd 2 960.3.c.h.449.3 4
120.59 even 2 960.3.c.g.449.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
60.3.b.a.29.1 ✓ 4 1.1 even 1 trivial
60.3.b.a.29.2 yes 4 15.14 odd 2 inner
60.3.b.a.29.3 yes 4 3.2 odd 2 inner
60.3.b.a.29.4 yes 4 5.4 even 2 inner
240.3.c.d.209.1 4 20.19 odd 2
240.3.c.d.209.2 4 12.11 even 2
240.3.c.d.209.3 4 60.59 even 2
240.3.c.d.209.4 4 4.3 odd 2
300.3.g.e.101.1 2 15.2 even 4
300.3.g.e.101.2 2 5.2 odd 4
300.3.g.h.101.1 2 5.3 odd 4
300.3.g.h.101.2 2 15.8 even 4
960.3.c.g.449.1 4 8.3 odd 2
960.3.c.g.449.2 4 120.59 even 2
960.3.c.g.449.3 4 24.11 even 2
960.3.c.g.449.4 4 40.19 odd 2
960.3.c.h.449.1 4 40.29 even 2
960.3.c.h.449.2 4 24.5 odd 2
960.3.c.h.449.3 4 120.29 odd 2
960.3.c.h.449.4 4 8.5 even 2
1200.3.l.h.401.1 2 60.23 odd 4
1200.3.l.h.401.2 2 20.3 even 4
1200.3.l.q.401.1 2 20.7 even 4
1200.3.l.q.401.2 2 60.47 odd 4
1620.3.t.b.269.1 8 45.4 even 6
1620.3.t.b.269.2 8 9.4 even 3
1620.3.t.b.269.3 8 9.5 odd 6
1620.3.t.b.269.4 8 45.14 odd 6
1620.3.t.b.1349.1 8 9.2 odd 6
1620.3.t.b.1349.2 8 45.29 odd 6
1620.3.t.b.1349.3 8 45.34 even 6
1620.3.t.b.1349.4 8 9.7 even 3