Properties

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

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Newspace parameters

Level: \( N \) \(=\) \( 60 = 2^{2} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 60.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.479102412128\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + i q^{3} + (2 i + 1) q^{5} - 4 i q^{7} - q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + i q^{3} + (2 i + 1) q^{5} - 4 i q^{7} - q^{9} - 4 q^{11} + (i - 2) q^{15} - 4 i q^{17} + 4 q^{21} + 4 i q^{23} + (4 i - 3) q^{25} - i q^{27} + 6 q^{29} + 4 q^{31} - 4 i q^{33} + ( - 4 i + 8) q^{35} + 8 i q^{37} - 10 q^{41} + 4 i q^{43} + ( - 2 i - 1) q^{45} + 4 i q^{47} - 9 q^{49} + 4 q^{51} - 12 i q^{53} + ( - 8 i - 4) q^{55} - 4 q^{59} + 2 q^{61} + 4 i q^{63} + 4 i q^{67} - 4 q^{69} - 8 i q^{73} + ( - 3 i - 4) q^{75} + 16 i q^{77} + 12 q^{79} + q^{81} + 4 i q^{83} + ( - 4 i + 8) q^{85} + 6 i q^{87} + 10 q^{89} + 4 i q^{93} - 8 i q^{97} + 4 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{5} - 2 q^{9} - 8 q^{11} - 4 q^{15} + 8 q^{21} - 6 q^{25} + 12 q^{29} + 8 q^{31} + 16 q^{35} - 20 q^{41} - 2 q^{45} - 18 q^{49} + 8 q^{51} - 8 q^{55} - 8 q^{59} + 4 q^{61} - 8 q^{69} - 8 q^{75} + 24 q^{79} + 2 q^{81} + 16 q^{85} + 20 q^{89} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
1.00000i
1.00000i
0 1.00000i 0 1.00000 2.00000i 0 4.00000i 0 −1.00000 0
49.2 0 1.00000i 0 1.00000 + 2.00000i 0 4.00000i 0 −1.00000 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
5.b 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.d.a 2
3.b odd 2 1 180.2.d.a 2
4.b odd 2 1 240.2.f.b 2
5.b even 2 1 inner 60.2.d.a 2
5.c odd 4 1 300.2.a.a 1
5.c odd 4 1 300.2.a.d 1
7.b odd 2 1 2940.2.k.c 2
7.c even 3 2 2940.2.bb.d 4
7.d odd 6 2 2940.2.bb.e 4
8.b even 2 1 960.2.f.f 2
8.d odd 2 1 960.2.f.c 2
9.c even 3 2 1620.2.r.c 4
9.d odd 6 2 1620.2.r.d 4
12.b even 2 1 720.2.f.c 2
15.d odd 2 1 180.2.d.a 2
15.e even 4 1 900.2.a.a 1
15.e even 4 1 900.2.a.h 1
16.e even 4 1 3840.2.d.o 2
16.e even 4 1 3840.2.d.r 2
16.f odd 4 1 3840.2.d.b 2
16.f odd 4 1 3840.2.d.be 2
20.d odd 2 1 240.2.f.b 2
20.e even 4 1 1200.2.a.a 1
20.e even 4 1 1200.2.a.s 1
24.f even 2 1 2880.2.f.p 2
24.h odd 2 1 2880.2.f.l 2
35.c odd 2 1 2940.2.k.c 2
35.i odd 6 2 2940.2.bb.e 4
35.j even 6 2 2940.2.bb.d 4
40.e odd 2 1 960.2.f.c 2
40.f even 2 1 960.2.f.f 2
40.i odd 4 1 4800.2.a.bj 1
40.i odd 4 1 4800.2.a.bn 1
40.k even 4 1 4800.2.a.bf 1
40.k even 4 1 4800.2.a.bk 1
45.h odd 6 2 1620.2.r.d 4
45.j even 6 2 1620.2.r.c 4
60.h even 2 1 720.2.f.c 2
60.l odd 4 1 3600.2.a.d 1
60.l odd 4 1 3600.2.a.bm 1
80.k odd 4 1 3840.2.d.b 2
80.k odd 4 1 3840.2.d.be 2
80.q even 4 1 3840.2.d.o 2
80.q even 4 1 3840.2.d.r 2
120.i odd 2 1 2880.2.f.l 2
120.m even 2 1 2880.2.f.p 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.2.d.a 2 1.a even 1 1 trivial
60.2.d.a 2 5.b even 2 1 inner
180.2.d.a 2 3.b odd 2 1
180.2.d.a 2 15.d odd 2 1
240.2.f.b 2 4.b odd 2 1
240.2.f.b 2 20.d odd 2 1
300.2.a.a 1 5.c odd 4 1
300.2.a.d 1 5.c odd 4 1
720.2.f.c 2 12.b even 2 1
720.2.f.c 2 60.h even 2 1
900.2.a.a 1 15.e even 4 1
900.2.a.h 1 15.e even 4 1
960.2.f.c 2 8.d odd 2 1
960.2.f.c 2 40.e odd 2 1
960.2.f.f 2 8.b even 2 1
960.2.f.f 2 40.f even 2 1
1200.2.a.a 1 20.e even 4 1
1200.2.a.s 1 20.e even 4 1
1620.2.r.c 4 9.c even 3 2
1620.2.r.c 4 45.j even 6 2
1620.2.r.d 4 9.d odd 6 2
1620.2.r.d 4 45.h odd 6 2
2880.2.f.l 2 24.h odd 2 1
2880.2.f.l 2 120.i odd 2 1
2880.2.f.p 2 24.f even 2 1
2880.2.f.p 2 120.m even 2 1
2940.2.k.c 2 7.b odd 2 1
2940.2.k.c 2 35.c odd 2 1
2940.2.bb.d 4 7.c even 3 2
2940.2.bb.d 4 35.j even 6 2
2940.2.bb.e 4 7.d odd 6 2
2940.2.bb.e 4 35.i odd 6 2
3600.2.a.d 1 60.l odd 4 1
3600.2.a.bm 1 60.l odd 4 1
3840.2.d.b 2 16.f odd 4 1
3840.2.d.b 2 80.k odd 4 1
3840.2.d.o 2 16.e even 4 1
3840.2.d.o 2 80.q even 4 1
3840.2.d.r 2 16.e even 4 1
3840.2.d.r 2 80.q even 4 1
3840.2.d.be 2 16.f odd 4 1
3840.2.d.be 2 80.k odd 4 1
4800.2.a.bf 1 40.k even 4 1
4800.2.a.bj 1 40.i odd 4 1
4800.2.a.bk 1 40.k even 4 1
4800.2.a.bn 1 40.i odd 4 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{2} - 2T + 5 \) Copy content Toggle raw display
$7$ \( T^{2} + 16 \) Copy content Toggle raw display
$11$ \( (T + 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 16 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 16 \) Copy content Toggle raw display
$29$ \( (T - 6)^{2} \) Copy content Toggle raw display
$31$ \( (T - 4)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 64 \) Copy content Toggle raw display
$41$ \( (T + 10)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 16 \) Copy content Toggle raw display
$47$ \( T^{2} + 16 \) Copy content Toggle raw display
$53$ \( T^{2} + 144 \) Copy content Toggle raw display
$59$ \( (T + 4)^{2} \) Copy content Toggle raw display
$61$ \( (T - 2)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 16 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 64 \) Copy content Toggle raw display
$79$ \( (T - 12)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 16 \) Copy content Toggle raw display
$89$ \( (T - 10)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 64 \) Copy content Toggle raw display
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