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Properties

Label 60.2.j.a
Level $60$
Weight $2$
Character orbit 60.j
Analytic conductor $0.479$
Analytic rank $0$
Dimension $12$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [60,2,Mod(7,60)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(60, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("60.7");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 60 = 2^{2} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 60.j (of order \(4\), degree \(2\), 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: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: 12.0.426337261060096.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4x^{9} - 3x^{8} + 4x^{7} + 8x^{6} + 8x^{5} - 12x^{4} - 32x^{3} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{6} q^{2} - \beta_{2} q^{3} + ( - \beta_{7} + \beta_{4} + \beta_{2}) q^{4} + ( - \beta_{8} + \beta_{6} + \cdots - \beta_{4}) q^{5}+ \cdots + \beta_{8} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{6} q^{2} - \beta_{2} q^{3} + ( - \beta_{7} + \beta_{4} + \beta_{2}) q^{4} + ( - \beta_{8} + \beta_{6} + \cdots - \beta_{4}) q^{5}+ \cdots + ( - \beta_{11} - \beta_{10} + \cdots + 2 \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 4 q^{6} - 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 4 q^{6} - 12 q^{8} - 8 q^{10} - 8 q^{12} - 4 q^{13} + 12 q^{16} - 20 q^{17} + 20 q^{20} + 12 q^{22} - 20 q^{25} + 16 q^{26} - 4 q^{28} + 8 q^{30} + 20 q^{32} + 8 q^{33} + 4 q^{36} + 4 q^{37} + 16 q^{38} - 8 q^{40} + 16 q^{41} + 20 q^{42} + 4 q^{45} - 40 q^{46} + 16 q^{48} - 16 q^{50} - 8 q^{52} + 4 q^{53} - 64 q^{56} - 20 q^{58} - 20 q^{60} - 32 q^{61} - 56 q^{62} + 20 q^{65} - 24 q^{66} - 16 q^{68} + 44 q^{70} - 12 q^{72} + 44 q^{73} + 8 q^{76} + 48 q^{77} - 24 q^{78} + 4 q^{80} - 12 q^{81} + 16 q^{82} + 44 q^{85} + 64 q^{86} + 60 q^{88} + 12 q^{90} + 56 q^{92} - 16 q^{93} + 44 q^{96} - 20 q^{97} + 24 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4x^{9} - 3x^{8} + 4x^{7} + 8x^{6} + 8x^{5} - 12x^{4} - 32x^{3} + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{10} - \nu^{9} - 4\nu^{8} + 2\nu^{7} + 3\nu^{6} - 5\nu^{5} - 24\nu^{4} - 2\nu^{3} + 8\nu^{2} + 24\nu + 48 ) / 80 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{11} + \nu^{10} + 4\nu^{9} - 2\nu^{8} - 3\nu^{7} + 5\nu^{6} + 24\nu^{5} + 2\nu^{4} - 8\nu^{3} - 24\nu^{2} - 48\nu ) / 160 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - \nu^{10} + 9 \nu^{9} + 6 \nu^{8} + 2 \nu^{7} - 17 \nu^{6} - 35 \nu^{5} + 26 \nu^{4} + 38 \nu^{3} + \cdots - 192 ) / 80 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - \nu^{11} - \nu^{10} + \nu^{9} + 2 \nu^{8} + 13 \nu^{7} - 5 \nu^{6} - 19 \nu^{5} - 22 \nu^{4} + \cdots + 88 \nu ) / 80 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{11} + 4\nu^{10} + 2\nu^{9} + \nu^{7} + 16\nu^{6} - 6\nu^{5} + 4\nu^{4} + 8\nu^{3} - 48\nu^{2} - 160\nu - 64 ) / 160 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{11} + 4\nu^{8} + 3\nu^{7} - 4\nu^{6} - 8\nu^{5} - 8\nu^{4} + 12\nu^{3} + 32\nu^{2} ) / 32 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - \nu^{11} + 9 \nu^{10} + 6 \nu^{9} + 2 \nu^{8} - 17 \nu^{7} - 35 \nu^{6} + 26 \nu^{5} + \cdots - 192 \nu ) / 160 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 3\nu^{11} + \nu^{10} - 4\nu^{8} - 5\nu^{7} + \nu^{6} + 12\nu^{5} + 8\nu^{4} + 12\nu^{3} - 36\nu^{2} - 16\nu - 64 ) / 80 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 9 \nu^{11} + 6 \nu^{10} - 2 \nu^{9} - 20 \nu^{8} - 31 \nu^{7} + 34 \nu^{6} + 46 \nu^{5} + 16 \nu^{4} + \cdots + 64 ) / 160 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - \nu^{11} - 3 \nu^{10} - \nu^{9} + 4 \nu^{8} + 7 \nu^{7} + \nu^{6} - 9 \nu^{5} - 20 \nu^{4} + \cdots + 16 ) / 40 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 11 \nu^{11} + 6 \nu^{10} + 14 \nu^{9} - 12 \nu^{8} - 53 \nu^{7} + 10 \nu^{6} + 14 \nu^{5} + 112 \nu^{4} + \cdots - 320 ) / 160 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{11} - \beta_{9} + \beta_{4} - \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{11} - 2\beta_{10} + \beta_{9} - 2\beta_{7} + 2\beta_{6} - 2\beta_{5} - \beta_{4} + \beta_{3} + 2\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{11} - 3\beta_{9} + 2\beta_{8} + 2\beta_{5} - \beta_{4} - \beta_{3} + 2\beta _1 + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -\beta_{11} - 2\beta_{10} - \beta_{9} - 2\beta_{7} - 2\beta_{6} - \beta_{4} + \beta_{3} - 2\beta_{2} - 4\beta _1 + 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -\beta_{11} - \beta_{9} + 2\beta_{8} - 2\beta_{5} - \beta_{4} - \beta_{3} + 8\beta_{2} - 2\beta _1 - 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 3 \beta_{11} - 2 \beta_{10} - 3 \beta_{9} - 4 \beta_{8} - 6 \beta_{7} + 2 \beta_{6} + 6 \beta_{5} + \cdots + 6 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -5\beta_{11} - 5\beta_{9} + 10\beta_{8} - 8\beta_{6} + 10\beta_{5} + 5\beta_{4} + \beta_{3} - 6\beta _1 + 10 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 5 \beta_{11} + 10 \beta_{10} + 5 \beta_{9} + 2 \beta_{7} + 10 \beta_{6} + 5 \beta_{4} - 5 \beta_{3} + \cdots + 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 9\beta_{11} - 7\beta_{9} - 10\beta_{8} + 2\beta_{5} + 9\beta_{4} + \beta_{3} + 40\beta_{2} + 10\beta _1 + 10 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 13 \beta_{11} - 14 \beta_{10} - 13 \beta_{9} + 4 \beta_{8} + 6 \beta_{7} + 14 \beta_{6} + \cdots - 6 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 11 \beta_{11} + 29 \beta_{9} + 54 \beta_{8} - 16 \beta_{7} + 24 \beta_{6} - 18 \beta_{5} + 11 \beta_{4} + \cdots + 54 ) / 2 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(31\) \(37\) \(41\)
\(\chi(n)\) \(-1\) \(\beta_{8}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
1.41127 + 0.0912546i
1.19252 + 0.760198i
−0.0912546 − 1.41127i
−0.394157 + 1.35818i
−0.760198 − 1.19252i
−1.35818 + 0.394157i
1.41127 − 0.0912546i
1.19252 − 0.760198i
−0.0912546 + 1.41127i
−0.394157 − 1.35818i
−0.760198 + 1.19252i
−1.35818 − 0.394157i
−1.41127 + 0.0912546i −0.707107 − 0.707107i 1.98335 − 0.257569i 1.32001 − 1.80487i 1.06244 + 0.933389i 1.86678 − 1.86678i −2.77552 + 0.544488i 1.00000i −1.69819 + 2.66761i
7.2 −1.19252 + 0.760198i 0.707107 + 0.707107i 0.844199 − 1.81310i 0.432320 + 2.19388i −1.38078 − 0.305697i −0.611393 + 0.611393i 0.371591 + 2.80391i 1.00000i −2.18333 − 2.28759i
7.3 0.0912546 − 1.41127i 0.707107 + 0.707107i −1.98335 − 0.257569i 1.32001 − 1.80487i 1.06244 − 0.933389i −1.86678 + 1.86678i −0.544488 + 2.77552i 1.00000i −2.42670 − 2.02759i
7.4 0.394157 + 1.35818i 0.707107 + 0.707107i −1.68928 + 1.07067i −1.75233 − 1.38900i −0.681664 + 1.23909i 2.47817 − 2.47817i −2.12000 − 1.87233i 1.00000i 1.19582 − 2.92746i
7.5 0.760198 − 1.19252i −0.707107 − 0.707107i −0.844199 − 1.81310i 0.432320 + 2.19388i −1.38078 + 0.305697i 0.611393 − 0.611393i −2.80391 − 0.371591i 1.00000i 2.94489 + 1.15223i
7.6 1.35818 + 0.394157i −0.707107 − 0.707107i 1.68928 + 1.07067i −1.75233 − 1.38900i −0.681664 − 1.23909i −2.47817 + 2.47817i 1.87233 + 2.12000i 1.00000i −1.83249 − 2.57720i
43.1 −1.41127 − 0.0912546i −0.707107 + 0.707107i 1.98335 + 0.257569i 1.32001 + 1.80487i 1.06244 − 0.933389i 1.86678 + 1.86678i −2.77552 − 0.544488i − 1.00000i −1.69819 − 2.66761i
43.2 −1.19252 − 0.760198i 0.707107 − 0.707107i 0.844199 + 1.81310i 0.432320 − 2.19388i −1.38078 + 0.305697i −0.611393 − 0.611393i 0.371591 − 2.80391i − 1.00000i −2.18333 + 2.28759i
43.3 0.0912546 + 1.41127i 0.707107 − 0.707107i −1.98335 + 0.257569i 1.32001 + 1.80487i 1.06244 + 0.933389i −1.86678 − 1.86678i −0.544488 − 2.77552i − 1.00000i −2.42670 + 2.02759i
43.4 0.394157 − 1.35818i 0.707107 − 0.707107i −1.68928 − 1.07067i −1.75233 + 1.38900i −0.681664 − 1.23909i 2.47817 + 2.47817i −2.12000 + 1.87233i − 1.00000i 1.19582 + 2.92746i
43.5 0.760198 + 1.19252i −0.707107 + 0.707107i −0.844199 + 1.81310i 0.432320 − 2.19388i −1.38078 − 0.305697i 0.611393 + 0.611393i −2.80391 + 0.371591i − 1.00000i 2.94489 − 1.15223i
43.6 1.35818 − 0.394157i −0.707107 + 0.707107i 1.68928 − 1.07067i −1.75233 + 1.38900i −0.681664 + 1.23909i −2.47817 − 2.47817i 1.87233 − 2.12000i − 1.00000i −1.83249 + 2.57720i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.6
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 60.2.j.a ✓ 12
3.b odd 2 1 180.2.k.e 12
4.b odd 2 1 inner 60.2.j.a ✓ 12
5.b even 2 1 300.2.j.d 12
5.c odd 4 1 inner 60.2.j.a ✓ 12
5.c odd 4 1 300.2.j.d 12
8.b even 2 1 960.2.w.g 12
8.d odd 2 1 960.2.w.g 12
12.b even 2 1 180.2.k.e 12
15.d odd 2 1 900.2.k.n 12
15.e even 4 1 180.2.k.e 12
15.e even 4 1 900.2.k.n 12
20.d odd 2 1 300.2.j.d 12
20.e even 4 1 inner 60.2.j.a ✓ 12
20.e even 4 1 300.2.j.d 12
40.i odd 4 1 960.2.w.g 12
40.k even 4 1 960.2.w.g 12
60.h even 2 1 900.2.k.n 12
60.l odd 4 1 180.2.k.e 12
60.l odd 4 1 900.2.k.n 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.2.j.a ✓ 12 1.a even 1 1 trivial
60.2.j.a ✓ 12 4.b odd 2 1 inner
60.2.j.a ✓ 12 5.c odd 4 1 inner
60.2.j.a ✓ 12 20.e even 4 1 inner
180.2.k.e 12 3.b odd 2 1
180.2.k.e 12 12.b even 2 1
180.2.k.e 12 15.e even 4 1
180.2.k.e 12 60.l odd 4 1
300.2.j.d 12 5.b even 2 1
300.2.j.d 12 5.c odd 4 1
300.2.j.d 12 20.d odd 2 1
300.2.j.d 12 20.e even 4 1
900.2.k.n 12 15.d odd 2 1
900.2.k.n 12 15.e even 4 1
900.2.k.n 12 60.h even 2 1
900.2.k.n 12 60.l odd 4 1
960.2.w.g 12 8.b even 2 1
960.2.w.g 12 8.d odd 2 1
960.2.w.g 12 40.i odd 4 1
960.2.w.g 12 40.k even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(60, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 4 T^{9} + \cdots + 64 \) Copy content Toggle raw display
$3$ \( (T^{4} + 1)^{3} \) Copy content Toggle raw display
$5$ \( (T^{6} + 5 T^{4} + \cdots + 125)^{2} \) Copy content Toggle raw display
$7$ \( T^{12} + 200 T^{8} + \cdots + 4096 \) Copy content Toggle raw display
$11$ \( (T^{6} + 36 T^{4} + \cdots + 128)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + 2 T^{5} + 2 T^{4} + \cdots + 32)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + 10 T^{5} + \cdots + 800)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} - 40 T^{4} + \cdots - 512)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + 4640 T^{8} + \cdots + 65536 \) Copy content Toggle raw display
$29$ \( (T^{6} + 20 T^{4} + \cdots + 64)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + 112 T^{4} + \cdots + 32768)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} - 2 T^{5} + 2 T^{4} + \cdots + 32)^{2} \) Copy content Toggle raw display
$41$ \( (T^{3} - 4 T^{2} - 20 T + 64)^{4} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 15352201216 \) Copy content Toggle raw display
$47$ \( T^{12} + 4896 T^{8} + \cdots + 40960000 \) Copy content Toggle raw display
$53$ \( (T^{6} - 2 T^{5} + \cdots + 128)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} - 100 T^{4} + \cdots - 512)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + 8 T^{2} + \cdots + 176)^{4} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 15352201216 \) Copy content Toggle raw display
$71$ \( (T^{6} + 256 T^{4} + \cdots + 204800)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} - 22 T^{5} + \cdots + 55112)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} - 304 T^{4} + \cdots - 2048)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + 16672 T^{8} + \cdots + 65536 \) Copy content Toggle raw display
$89$ \( (T^{6} + 72 T^{4} + \cdots + 1024)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + 10 T^{5} + \cdots + 35912)^{2} \) Copy content Toggle raw display
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