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Properties

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

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [60,2,Mod(59,60)]
 
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, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//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);
 
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

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: \(0.479102412128\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-5})\)
comment: defining polynomial
 
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: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} - \beta_1 q^{3} - 2 q^{4} - \beta_{3} q^{5} + (\beta_{3} - 1) q^{6} + ( - \beta_{2} + 2 \beta_1) q^{7} + 2 \beta_{2} q^{8} + (\beta_{3} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} - \beta_1 q^{3} - 2 q^{4} - \beta_{3} q^{5} + (\beta_{3} - 1) q^{6} + ( - \beta_{2} + 2 \beta_1) q^{7} + 2 \beta_{2} q^{8} + (\beta_{3} + 2) q^{9} + (\beta_{2} - 2 \beta_1) q^{10} + 2 \beta_1 q^{12} - 2 \beta_{3} q^{14} + (3 \beta_{2} - \beta_1) q^{15} + 4 q^{16} + ( - 3 \beta_{2} + 2 \beta_1) q^{18} + 2 \beta_{3} q^{20} + ( - \beta_{3} - 5) q^{21} + \beta_{2} q^{23} + ( - 2 \beta_{3} + 2) q^{24} - 5 q^{25} + ( - 3 \beta_{2} - \beta_1) q^{27} + (2 \beta_{2} - 4 \beta_1) q^{28} + 4 \beta_{3} q^{29} + (\beta_{3} + 5) q^{30} - 4 \beta_{2} q^{32} - 5 \beta_{2} q^{35} + ( - 2 \beta_{3} - 4) q^{36} + ( - 2 \beta_{2} + 4 \beta_1) q^{40} - 2 \beta_{3} q^{41} + (6 \beta_{2} - 2 \beta_1) q^{42} + ( - \beta_{2} + 2 \beta_1) q^{43} + ( - 2 \beta_{3} + 5) q^{45} + 2 q^{46} + 7 \beta_{2} q^{47} - 4 \beta_1 q^{48} + 3 q^{49} + 5 \beta_{2} q^{50} + (\beta_{3} - 7) q^{54} + 4 \beta_{3} q^{56} + ( - 4 \beta_{2} + 8 \beta_1) q^{58} + ( - 6 \beta_{2} + 2 \beta_1) q^{60} + 8 q^{61} + (3 \beta_{2} + 4 \beta_1) q^{63} - 8 q^{64} + (5 \beta_{2} - 10 \beta_1) q^{67} + ( - \beta_{3} + 1) q^{69} - 10 q^{70} + (6 \beta_{2} - 4 \beta_1) q^{72} + 5 \beta_1 q^{75} - 4 \beta_{3} q^{80} + (4 \beta_{3} - 1) q^{81} + (2 \beta_{2} - 4 \beta_1) q^{82} - 11 \beta_{2} q^{83} + (2 \beta_{3} + 10) q^{84} - 2 \beta_{3} q^{86} + ( - 12 \beta_{2} + 4 \beta_1) q^{87} - 8 \beta_{3} q^{89} + ( - 3 \beta_{2} - 4 \beta_1) q^{90} - 2 \beta_{2} q^{92} + 14 q^{94} + (4 \beta_{3} - 4) q^{96} - 3 \beta_{2} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{4} - 4 q^{6} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{4} - 4 q^{6} + 8 q^{9} + 16 q^{16} - 20 q^{21} + 8 q^{24} - 20 q^{25} + 20 q^{30} - 16 q^{36} + 20 q^{45} + 8 q^{46} + 12 q^{49} - 28 q^{54} + 32 q^{61} - 32 q^{64} + 4 q^{69} - 40 q^{70} - 4 q^{81} + 40 q^{84} + 56 q^{94} - 16 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 4x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - \nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{2} + \beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
59.1
1.58114 + 0.707107i
−1.58114 + 0.707107i
1.58114 − 0.707107i
−1.58114 − 0.707107i
− 1.41421i −1.58114 − 0.707107i −2.00000 − 2.23607i −1.00000 + 2.23607i 3.16228 2.82843i 2.00000 + 2.23607i −3.16228
59.2 − 1.41421i 1.58114 − 0.707107i −2.00000 2.23607i −1.00000 − 2.23607i −3.16228 2.82843i 2.00000 − 2.23607i 3.16228
59.3 1.41421i −1.58114 + 0.707107i −2.00000 2.23607i −1.00000 − 2.23607i 3.16228 − 2.82843i 2.00000 − 2.23607i −3.16228
59.4 1.41421i 1.58114 + 0.707107i −2.00000 − 2.23607i −1.00000 + 2.23607i −3.16228 − 2.82843i 2.00000 + 2.23607i 3.16228
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)
3.b odd 2 1 inner
4.b odd 2 1 inner
5.b even 2 1 inner
12.b even 2 1 inner
15.d odd 2 1 inner
60.h even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 60.2.h.a ✓ 4
3.b odd 2 1 inner 60.2.h.a ✓ 4
4.b odd 2 1 inner 60.2.h.a ✓ 4
5.b even 2 1 inner 60.2.h.a ✓ 4
5.c odd 4 2 300.2.e.b 4
8.b even 2 1 960.2.o.c 4
8.d odd 2 1 960.2.o.c 4
12.b even 2 1 inner 60.2.h.a ✓ 4
15.d odd 2 1 inner 60.2.h.a ✓ 4
15.e even 4 2 300.2.e.b 4
20.d odd 2 1 CM 60.2.h.a ✓ 4
20.e even 4 2 300.2.e.b 4
24.f even 2 1 960.2.o.c 4
24.h odd 2 1 960.2.o.c 4
40.e odd 2 1 960.2.o.c 4
40.f even 2 1 960.2.o.c 4
60.h even 2 1 inner 60.2.h.a ✓ 4
60.l odd 4 2 300.2.e.b 4
120.i odd 2 1 960.2.o.c 4
120.m even 2 1 960.2.o.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.2.h.a ✓ 4 1.a even 1 1 trivial
60.2.h.a ✓ 4 3.b odd 2 1 inner
60.2.h.a ✓ 4 4.b odd 2 1 inner
60.2.h.a ✓ 4 5.b even 2 1 inner
60.2.h.a ✓ 4 12.b even 2 1 inner
60.2.h.a ✓ 4 15.d odd 2 1 inner
60.2.h.a ✓ 4 20.d odd 2 1 CM
60.2.h.a ✓ 4 60.h even 2 1 inner
300.2.e.b 4 5.c odd 4 2
300.2.e.b 4 15.e even 4 2
300.2.e.b 4 20.e even 4 2
300.2.e.b 4 60.l odd 4 2
960.2.o.c 4 8.b even 2 1
960.2.o.c 4 8.d odd 2 1
960.2.o.c 4 24.f even 2 1
960.2.o.c 4 24.h odd 2 1
960.2.o.c 4 40.e odd 2 1
960.2.o.c 4 40.f even 2 1
960.2.o.c 4 120.i odd 2 1
960.2.o.c 4 120.m even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} - 10 \) acting on \(S_{2}^{\mathrm{new}}(60, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} - 4T^{2} + 9 \) Copy content Toggle raw display
$5$ \( (T^{2} + 5)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} - 10)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 80)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} + 20)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 10)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 98)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( (T - 8)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} - 250)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 242)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 320)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} \) Copy content Toggle raw display
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