Properties

Label 60.2.i.a
Level $60$
Weight $2$
Character orbit 60.i
Analytic conductor $0.479$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 60 = 2^{2} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 60.i (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.479102412128\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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.

\(f(q)\) \(=\) \( q + (\beta_1 + 1) q^{3} + (\beta_{3} - \beta_1 - 1) q^{5} + ( - \beta_{2} - 1) q^{7} + ( - \beta_{3} + 2 \beta_{2} + \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 + 1) q^{3} + (\beta_{3} - \beta_1 - 1) q^{5} + ( - \beta_{2} - 1) q^{7} + ( - \beta_{3} + 2 \beta_{2} + \beta_1 + 1) q^{9} + ( - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{11} + (3 \beta_{2} - 3) q^{13} + (\beta_{3} - 2 \beta_{2} - 3) q^{15} + (2 \beta_{3} + \beta_{2} - 1) q^{17} - 2 \beta_{2} q^{19} + ( - \beta_{3} - \beta_{2} - \beta_1 - 1) q^{21} + (\beta_{2} + 2 \beta_1 + 1) q^{23} + 5 q^{25} + (\beta_{3} + 4 \beta_{2} + 3) q^{27} + ( - 2 \beta_{3} + 2 \beta_1 + 2) q^{29} + 4 q^{31} + ( - 6 \beta_{2} - 2 \beta_1 + 4) q^{33} + (\beta_{2} + 2 \beta_1 + 1) q^{35} + ( - 3 \beta_{2} - 3) q^{37} + (3 \beta_{3} + 3 \beta_{2} - 3 \beta_1 - 3) q^{39} + (4 \beta_{3} + 4 \beta_{2} + 4 \beta_1) q^{41} + (3 \beta_{2} - 3) q^{43} + ( - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 5) q^{45} + ( - 6 \beta_{3} - 3 \beta_{2} + 3) q^{47} - 5 \beta_{2} q^{49} + (\beta_{3} + \beta_{2} + \beta_1 - 5) q^{51} + ( - \beta_{2} - 2 \beta_1 - 1) q^{53} + 10 \beta_{2} q^{55} + ( - 2 \beta_{3} - 2 \beta_{2}) q^{57} + (4 \beta_{3} - 4 \beta_1 - 4) q^{59} - 6 q^{61} + ( - 3 \beta_{2} - 2 \beta_1 + 1) q^{63} + ( - 6 \beta_{3} - 3 \beta_{2} + 3) q^{65} + ( - \beta_{2} - 1) q^{67} + ( - \beta_{3} + 5 \beta_{2} + \beta_1 + 1) q^{69} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{71} + ( - \beta_{2} + 1) q^{73} + (5 \beta_1 + 5) q^{75} + (4 \beta_{3} + 2 \beta_{2} - 2) q^{77} - 6 \beta_{2} q^{79} + (4 \beta_{3} + 4 \beta_{2} + 4 \beta_1 + 1) q^{81} + ( - 3 \beta_{2} - 6 \beta_1 - 3) q^{83} + ( - 5 \beta_{2} + 5) q^{85} + ( - 2 \beta_{3} + 4 \beta_{2} + 6) q^{87} + (2 \beta_{3} - 2 \beta_1 - 2) q^{89} + 6 q^{91} + (4 \beta_1 + 4) q^{93} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{95} + (9 \beta_{2} + 9) q^{97} + ( - 4 \beta_{3} - 10 \beta_{2} + 4 \beta_1 + 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{3} - 4 q^{7} - 12 q^{13} - 10 q^{15} - 4 q^{21} + 20 q^{25} + 14 q^{27} + 16 q^{31} + 20 q^{33} - 12 q^{37} - 12 q^{43} - 20 q^{45} - 20 q^{51} - 4 q^{57} - 24 q^{61} + 8 q^{63} - 4 q^{67} + 4 q^{73} + 10 q^{75} + 4 q^{81} + 20 q^{85} + 20 q^{87} + 24 q^{91} + 8 q^{93} + 36 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 3x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} + \nu + 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} + 2\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{2} + \nu - 1 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{3} + \beta _1 - 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{3} + \beta_{2} - \beta_1 \) Copy content Toggle raw display

Character values

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

\(n\) \(31\) \(37\) \(41\)
\(\chi(n)\) \(1\) \(\beta_{2}\) \(-1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
1.61803i
0.618034i
1.61803i
0.618034i
0 −0.618034 1.61803i 0 2.23607 0 −1.00000 1.00000i 0 −2.23607 + 2.00000i 0
17.2 0 1.61803 + 0.618034i 0 −2.23607 0 −1.00000 1.00000i 0 2.23607 + 2.00000i 0
53.1 0 −0.618034 + 1.61803i 0 2.23607 0 −1.00000 + 1.00000i 0 −2.23607 2.00000i 0
53.2 0 1.61803 0.618034i 0 −2.23607 0 −1.00000 + 1.00000i 0 2.23607 2.00000i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.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.i.a 4
3.b odd 2 1 inner 60.2.i.a 4
4.b odd 2 1 240.2.v.b 4
5.b even 2 1 300.2.i.a 4
5.c odd 4 1 inner 60.2.i.a 4
5.c odd 4 1 300.2.i.a 4
8.b even 2 1 960.2.v.e 4
8.d odd 2 1 960.2.v.h 4
9.c even 3 2 1620.2.x.b 8
9.d odd 6 2 1620.2.x.b 8
12.b even 2 1 240.2.v.b 4
15.d odd 2 1 300.2.i.a 4
15.e even 4 1 inner 60.2.i.a 4
15.e even 4 1 300.2.i.a 4
20.d odd 2 1 1200.2.v.i 4
20.e even 4 1 240.2.v.b 4
20.e even 4 1 1200.2.v.i 4
24.f even 2 1 960.2.v.h 4
24.h odd 2 1 960.2.v.e 4
40.i odd 4 1 960.2.v.e 4
40.k even 4 1 960.2.v.h 4
45.k odd 12 2 1620.2.x.b 8
45.l even 12 2 1620.2.x.b 8
60.h even 2 1 1200.2.v.i 4
60.l odd 4 1 240.2.v.b 4
60.l odd 4 1 1200.2.v.i 4
120.q odd 4 1 960.2.v.h 4
120.w even 4 1 960.2.v.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.2.i.a 4 1.a even 1 1 trivial
60.2.i.a 4 3.b odd 2 1 inner
60.2.i.a 4 5.c odd 4 1 inner
60.2.i.a 4 15.e even 4 1 inner
240.2.v.b 4 4.b odd 2 1
240.2.v.b 4 12.b even 2 1
240.2.v.b 4 20.e even 4 1
240.2.v.b 4 60.l odd 4 1
300.2.i.a 4 5.b even 2 1
300.2.i.a 4 5.c odd 4 1
300.2.i.a 4 15.d odd 2 1
300.2.i.a 4 15.e even 4 1
960.2.v.e 4 8.b even 2 1
960.2.v.e 4 24.h odd 2 1
960.2.v.e 4 40.i odd 4 1
960.2.v.e 4 120.w even 4 1
960.2.v.h 4 8.d odd 2 1
960.2.v.h 4 24.f even 2 1
960.2.v.h 4 40.k even 4 1
960.2.v.h 4 120.q odd 4 1
1200.2.v.i 4 20.d odd 2 1
1200.2.v.i 4 20.e even 4 1
1200.2.v.i 4 60.h even 2 1
1200.2.v.i 4 60.l odd 4 1
1620.2.x.b 8 9.c even 3 2
1620.2.x.b 8 9.d odd 6 2
1620.2.x.b 8 45.k odd 12 2
1620.2.x.b 8 45.l even 12 2

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 2 T^{3} + 2 T^{2} - 6 T + 9 \) Copy content Toggle raw display
$5$ \( (T^{2} - 5)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 2 T + 2)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 20)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 6 T + 18)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 100 \) Copy content Toggle raw display
$19$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 100 \) Copy content Toggle raw display
$29$ \( (T^{2} - 20)^{2} \) Copy content Toggle raw display
$31$ \( (T - 4)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 6 T + 18)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 80)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 6 T + 18)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 8100 \) Copy content Toggle raw display
$53$ \( T^{4} + 100 \) Copy content Toggle raw display
$59$ \( (T^{2} - 80)^{2} \) Copy content Toggle raw display
$61$ \( (T + 6)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 2 T + 2)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 20)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 2 T + 2)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 8100 \) Copy content Toggle raw display
$89$ \( (T^{2} - 20)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 18 T + 162)^{2} \) Copy content Toggle raw display
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