Properties

Label 60.4.d.a
Level $60$
Weight $4$
Character orbit 60.d
Analytic conductor $3.540$
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: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 60.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.54011460034\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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.

\(f(q)\) \(=\) \( q + 3 i q^{3} + ( -10 + 5 i ) q^{5} + 22 i q^{7} -9 q^{9} +O(q^{10})\) \( q + 3 i q^{3} + ( -10 + 5 i ) q^{5} + 22 i q^{7} -9 q^{9} -14 q^{11} + 30 i q^{13} + ( -15 - 30 i ) q^{15} + 62 i q^{17} + 120 q^{19} -66 q^{21} -188 i q^{23} + ( 75 - 100 i ) q^{25} -27 i q^{27} -96 q^{29} + 184 q^{31} -42 i q^{33} + ( -110 - 220 i ) q^{35} + 406 i q^{37} -90 q^{39} + 130 q^{41} -148 i q^{43} + ( 90 - 45 i ) q^{45} + 448 i q^{47} -141 q^{49} -186 q^{51} + 414 i q^{53} + ( 140 - 70 i ) q^{55} + 360 i q^{57} -266 q^{59} -838 q^{61} -198 i q^{63} + ( -150 - 300 i ) q^{65} + 248 i q^{67} + 564 q^{69} + 1020 q^{71} -484 i q^{73} + ( 300 + 225 i ) q^{75} -308 i q^{77} + 48 q^{79} + 81 q^{81} -548 i q^{83} + ( -310 - 620 i ) q^{85} -288 i q^{87} + 650 q^{89} -660 q^{91} + 552 i q^{93} + ( -1200 + 600 i ) q^{95} -1816 i q^{97} + 126 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 20q^{5} - 18q^{9} + O(q^{10}) \) \( 2q - 20q^{5} - 18q^{9} - 28q^{11} - 30q^{15} + 240q^{19} - 132q^{21} + 150q^{25} - 192q^{29} + 368q^{31} - 220q^{35} - 180q^{39} + 260q^{41} + 180q^{45} - 282q^{49} - 372q^{51} + 280q^{55} - 532q^{59} - 1676q^{61} - 300q^{65} + 1128q^{69} + 2040q^{71} + 600q^{75} + 96q^{79} + 162q^{81} - 620q^{85} + 1300q^{89} - 1320q^{91} - 2400q^{95} + 252q^{99} + O(q^{100}) \)

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 3.00000i 0 −10.0000 5.00000i 0 22.0000i 0 −9.00000 0
49.2 0 3.00000i 0 −10.0000 + 5.00000i 0 22.0000i 0 −9.00000 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 60.4.d.a 2
3.b odd 2 1 180.4.d.b 2
4.b odd 2 1 240.4.f.a 2
5.b even 2 1 inner 60.4.d.a 2
5.c odd 4 1 300.4.a.d 1
5.c odd 4 1 300.4.a.f 1
8.b even 2 1 960.4.f.j 2
8.d odd 2 1 960.4.f.i 2
12.b even 2 1 720.4.f.h 2
15.d odd 2 1 180.4.d.b 2
15.e even 4 1 900.4.a.d 1
15.e even 4 1 900.4.a.o 1
20.d odd 2 1 240.4.f.a 2
20.e even 4 1 1200.4.a.q 1
20.e even 4 1 1200.4.a.w 1
40.e odd 2 1 960.4.f.i 2
40.f even 2 1 960.4.f.j 2
60.h even 2 1 720.4.f.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.4.d.a 2 1.a even 1 1 trivial
60.4.d.a 2 5.b even 2 1 inner
180.4.d.b 2 3.b odd 2 1
180.4.d.b 2 15.d odd 2 1
240.4.f.a 2 4.b odd 2 1
240.4.f.a 2 20.d odd 2 1
300.4.a.d 1 5.c odd 4 1
300.4.a.f 1 5.c odd 4 1
720.4.f.h 2 12.b even 2 1
720.4.f.h 2 60.h even 2 1
900.4.a.d 1 15.e even 4 1
900.4.a.o 1 15.e even 4 1
960.4.f.i 2 8.d odd 2 1
960.4.f.i 2 40.e odd 2 1
960.4.f.j 2 8.b even 2 1
960.4.f.j 2 40.f even 2 1
1200.4.a.q 1 20.e even 4 1
1200.4.a.w 1 20.e even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(60, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 9 + T^{2} \)
$5$ \( 125 + 20 T + T^{2} \)
$7$ \( 484 + T^{2} \)
$11$ \( ( 14 + T )^{2} \)
$13$ \( 900 + T^{2} \)
$17$ \( 3844 + T^{2} \)
$19$ \( ( -120 + T )^{2} \)
$23$ \( 35344 + T^{2} \)
$29$ \( ( 96 + T )^{2} \)
$31$ \( ( -184 + T )^{2} \)
$37$ \( 164836 + T^{2} \)
$41$ \( ( -130 + T )^{2} \)
$43$ \( 21904 + T^{2} \)
$47$ \( 200704 + T^{2} \)
$53$ \( 171396 + T^{2} \)
$59$ \( ( 266 + T )^{2} \)
$61$ \( ( 838 + T )^{2} \)
$67$ \( 61504 + T^{2} \)
$71$ \( ( -1020 + T )^{2} \)
$73$ \( 234256 + T^{2} \)
$79$ \( ( -48 + T )^{2} \)
$83$ \( 300304 + T^{2} \)
$89$ \( ( -650 + T )^{2} \)
$97$ \( 3297856 + T^{2} \)
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