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Properties

Label 60.3.f.a
Level $60$
Weight $3$
Character orbit 60.f
Analytic conductor $1.635$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [60,3,Mod(19,60)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(60, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("60.19");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 60 = 2^{2} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 60.f (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.63488158616\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{3} + \beta_1) q^{2} + \beta_{3} q^{3} + (2 \beta_{2} + 2) q^{4} - 5 \beta_1 q^{5} + (\beta_{2} + 3) q^{6} - 6 \beta_{3} q^{7} + 8 \beta_1 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} + \beta_1) q^{2} + \beta_{3} q^{3} + (2 \beta_{2} + 2) q^{4} - 5 \beta_1 q^{5} + (\beta_{2} + 3) q^{6} - 6 \beta_{3} q^{7} + 8 \beta_1 q^{8} + 3 q^{9} + ( - 5 \beta_{2} + 5) q^{10} + 6 \beta_{2} q^{11} + (2 \beta_{3} + 6 \beta_1) q^{12} - 18 \beta_1 q^{13} + ( - 6 \beta_{2} - 18) q^{14} - 5 \beta_{2} q^{15} + (8 \beta_{2} - 8) q^{16} + 10 \beta_1 q^{17} + (3 \beta_{3} + 3 \beta_1) q^{18} - 8 \beta_{2} q^{19} + (10 \beta_{3} - 10 \beta_1) q^{20} - 18 q^{21} + ( - 6 \beta_{3} + 18 \beta_1) q^{22} + 4 \beta_{3} q^{23} + 8 \beta_{2} q^{24} - 25 q^{25} + ( - 18 \beta_{2} + 18) q^{26} + 3 \beta_{3} q^{27} + ( - 12 \beta_{3} - 36 \beta_1) q^{28} + 36 q^{29} + (5 \beta_{3} - 15 \beta_1) q^{30} - 4 \beta_{2} q^{31} + ( - 16 \beta_{3} + 16 \beta_1) q^{32} + 18 \beta_1 q^{33} + (10 \beta_{2} - 10) q^{34} + 30 \beta_{2} q^{35} + (6 \beta_{2} + 6) q^{36} + 54 \beta_1 q^{37} + (8 \beta_{3} - 24 \beta_1) q^{38} - 18 \beta_{2} q^{39} + 40 q^{40} + 18 q^{41} + ( - 18 \beta_{3} - 18 \beta_1) q^{42} + 12 \beta_{3} q^{43} + (12 \beta_{2} - 36) q^{44} - 15 \beta_1 q^{45} + (4 \beta_{2} + 12) q^{46} + ( - 8 \beta_{3} + 24 \beta_1) q^{48} + 59 q^{49} + ( - 25 \beta_{3} - 25 \beta_1) q^{50} + 10 \beta_{2} q^{51} + (36 \beta_{3} - 36 \beta_1) q^{52} + 26 \beta_1 q^{53} + (3 \beta_{2} + 9) q^{54} + 30 \beta_{3} q^{55} - 48 \beta_{2} q^{56} - 24 \beta_1 q^{57} + (36 \beta_{3} + 36 \beta_1) q^{58} + 18 \beta_{2} q^{59} + ( - 10 \beta_{2} + 30) q^{60} - 74 q^{61} + (4 \beta_{3} - 12 \beta_1) q^{62} - 18 \beta_{3} q^{63} - 64 q^{64} - 90 q^{65} + (18 \beta_{2} - 18) q^{66} - 24 \beta_{3} q^{67} + ( - 20 \beta_{3} + 20 \beta_1) q^{68} + 12 q^{69} + ( - 30 \beta_{3} + 90 \beta_1) q^{70} - 60 \beta_{2} q^{71} + 24 \beta_1 q^{72} - 36 \beta_1 q^{73} + (54 \beta_{2} - 54) q^{74} - 25 \beta_{3} q^{75} + ( - 16 \beta_{2} + 48) q^{76} - 108 \beta_1 q^{77} + (18 \beta_{3} - 54 \beta_1) q^{78} - 52 \beta_{2} q^{79} + (40 \beta_{3} + 40 \beta_1) q^{80} + 9 q^{81} + (18 \beta_{3} + 18 \beta_1) q^{82} - 52 \beta_{3} q^{83} + ( - 36 \beta_{2} - 36) q^{84} + 50 q^{85} + (12 \beta_{2} + 36) q^{86} + 36 \beta_{3} q^{87} - 48 \beta_{3} q^{88} + 18 q^{89} + ( - 15 \beta_{2} + 15) q^{90} + 108 \beta_{2} q^{91} + (8 \beta_{3} + 24 \beta_1) q^{92} - 12 \beta_1 q^{93} - 40 \beta_{3} q^{95} + (16 \beta_{2} - 48) q^{96} - 72 \beta_1 q^{97} + (59 \beta_{3} + 59 \beta_1) q^{98} + 18 \beta_{2} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{4} + 12 q^{6} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 8 q^{4} + 12 q^{6} + 12 q^{9} + 20 q^{10} - 72 q^{14} - 32 q^{16} - 72 q^{21} - 100 q^{25} + 72 q^{26} + 144 q^{29} - 40 q^{34} + 24 q^{36} + 160 q^{40} + 72 q^{41} - 144 q^{44} + 48 q^{46} + 236 q^{49} + 36 q^{54} + 120 q^{60} - 296 q^{61} - 256 q^{64} - 360 q^{65} - 72 q^{66} + 48 q^{69} - 216 q^{74} + 192 q^{76} + 36 q^{81} - 144 q^{84} + 200 q^{85} + 144 q^{86} + 72 q^{89} + 60 q^{90} - 192 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{12}^{3} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\zeta_{12}^{2} - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{12}^{3} + 2\zeta_{12} \) Copy content Toggle raw display

Display \(\zeta_{12}^j\) in terms of \(\beta_i\)

\(\zeta_{12}\)\(=\) \( ( \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( ( \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( \beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{12}^j\)

Character values

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

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

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
−0.866025 − 0.500000i
−0.866025 + 0.500000i
0.866025 − 0.500000i
0.866025 + 0.500000i
−1.73205 − 1.00000i −1.73205 2.00000 + 3.46410i 5.00000i 3.00000 + 1.73205i 10.3923 − 8.00000i 3.00000 5.00000 − 8.66025i
19.2 −1.73205 + 1.00000i −1.73205 2.00000 − 3.46410i − 5.00000i 3.00000 − 1.73205i 10.3923 8.00000i 3.00000 5.00000 + 8.66025i
19.3 1.73205 − 1.00000i 1.73205 2.00000 − 3.46410i 5.00000i 3.00000 − 1.73205i −10.3923 − 8.00000i 3.00000 5.00000 + 8.66025i
19.4 1.73205 + 1.00000i 1.73205 2.00000 + 3.46410i − 5.00000i 3.00000 + 1.73205i −10.3923 8.00000i 3.00000 5.00000 − 8.66025i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
20.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 60.3.f.a ✓ 4
3.b odd 2 1 180.3.f.e 4
4.b odd 2 1 inner 60.3.f.a ✓ 4
5.b even 2 1 inner 60.3.f.a ✓ 4
5.c odd 4 1 300.3.c.a 2
5.c odd 4 1 300.3.c.c 2
8.b even 2 1 960.3.j.b 4
8.d odd 2 1 960.3.j.b 4
12.b even 2 1 180.3.f.e 4
15.d odd 2 1 180.3.f.e 4
15.e even 4 1 900.3.c.f 2
15.e even 4 1 900.3.c.j 2
20.d odd 2 1 inner 60.3.f.a ✓ 4
20.e even 4 1 300.3.c.a 2
20.e even 4 1 300.3.c.c 2
40.e odd 2 1 960.3.j.b 4
40.f even 2 1 960.3.j.b 4
60.h even 2 1 180.3.f.e 4
60.l odd 4 1 900.3.c.f 2
60.l odd 4 1 900.3.c.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.3.f.a ✓ 4 1.a even 1 1 trivial
60.3.f.a ✓ 4 4.b odd 2 1 inner
60.3.f.a ✓ 4 5.b even 2 1 inner
60.3.f.a ✓ 4 20.d odd 2 1 inner
180.3.f.e 4 3.b odd 2 1
180.3.f.e 4 12.b even 2 1
180.3.f.e 4 15.d odd 2 1
180.3.f.e 4 60.h even 2 1
300.3.c.a 2 5.c odd 4 1
300.3.c.a 2 20.e even 4 1
300.3.c.c 2 5.c odd 4 1
300.3.c.c 2 20.e even 4 1
900.3.c.f 2 15.e even 4 1
900.3.c.f 2 60.l odd 4 1
900.3.c.j 2 15.e even 4 1
900.3.c.j 2 60.l odd 4 1
960.3.j.b 4 8.b even 2 1
960.3.j.b 4 8.d odd 2 1
960.3.j.b 4 40.e odd 2 1
960.3.j.b 4 40.f even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} - 108 \) acting on \(S_{3}^{\mathrm{new}}(60, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$3$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 25)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} - 108)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 108)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 324)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 100)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 192)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 48)^{2} \) Copy content Toggle raw display
$29$ \( (T - 36)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 48)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 2916)^{2} \) Copy content Toggle raw display
$41$ \( (T - 18)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} - 432)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} + 676)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 972)^{2} \) Copy content Toggle raw display
$61$ \( (T + 74)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} - 1728)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 10800)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 1296)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 8112)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 8112)^{2} \) Copy content Toggle raw display
$89$ \( (T - 18)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + 5184)^{2} \) Copy content Toggle raw display
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