Properties

Label 60.5.g.a
Level $60$
Weight $5$
Character orbit 60.g
Analytic conductor $6.202$
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: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 60.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.20219778503\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-5}) \)
Defining polynomial: \(x^{2} + 5\)
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 \(\beta = \sqrt{-5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 6 + 3 \beta ) q^{3} -5 \beta q^{5} + 74 q^{7} + ( -9 + 36 \beta ) q^{9} +O(q^{10})\) \( q + ( 6 + 3 \beta ) q^{3} -5 \beta q^{5} + 74 q^{7} + ( -9 + 36 \beta ) q^{9} + 54 \beta q^{11} -16 q^{13} + ( 75 - 30 \beta ) q^{15} + 78 \beta q^{17} + 374 q^{19} + ( 444 + 222 \beta ) q^{21} -294 \beta q^{23} -125 q^{25} + ( -594 + 189 \beta ) q^{27} -618 \beta q^{29} -1426 q^{31} + ( -810 + 324 \beta ) q^{33} -370 \beta q^{35} + 272 q^{37} + ( -96 - 48 \beta ) q^{39} -348 \beta q^{41} -1936 q^{43} + ( 900 + 45 \beta ) q^{45} -1434 \beta q^{47} + 3075 q^{49} + ( -1170 + 468 \beta ) q^{51} + 2454 \beta q^{53} + 1350 q^{55} + ( 2244 + 1122 \beta ) q^{57} -2154 \beta q^{59} + 2234 q^{61} + ( -666 + 2664 \beta ) q^{63} + 80 \beta q^{65} -5416 q^{67} + ( 4410 - 1764 \beta ) q^{69} + 696 \beta q^{71} -538 q^{73} + ( -750 - 375 \beta ) q^{75} + 3996 \beta q^{77} + 9962 q^{79} + ( -6399 - 648 \beta ) q^{81} + 2082 \beta q^{83} + 1950 q^{85} + ( 9270 - 3708 \beta ) q^{87} -2352 \beta q^{89} -1184 q^{91} + ( -8556 - 4278 \beta ) q^{93} -1870 \beta q^{95} -10726 q^{97} + ( -9720 - 486 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 12q^{3} + 148q^{7} - 18q^{9} + O(q^{10}) \) \( 2q + 12q^{3} + 148q^{7} - 18q^{9} - 32q^{13} + 150q^{15} + 748q^{19} + 888q^{21} - 250q^{25} - 1188q^{27} - 2852q^{31} - 1620q^{33} + 544q^{37} - 192q^{39} - 3872q^{43} + 1800q^{45} + 6150q^{49} - 2340q^{51} + 2700q^{55} + 4488q^{57} + 4468q^{61} - 1332q^{63} - 10832q^{67} + 8820q^{69} - 1076q^{73} - 1500q^{75} + 19924q^{79} - 12798q^{81} + 3900q^{85} + 18540q^{87} - 2368q^{91} - 17112q^{93} - 21452q^{97} - 19440q^{99} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
41.1
2.23607i
2.23607i
0 6.00000 6.70820i 0 11.1803i 0 74.0000 0 −9.00000 80.4984i 0
41.2 0 6.00000 + 6.70820i 0 11.1803i 0 74.0000 0 −9.00000 + 80.4984i 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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 60.5.g.a 2
3.b odd 2 1 inner 60.5.g.a 2
4.b odd 2 1 240.5.l.a 2
5.b even 2 1 300.5.g.d 2
5.c odd 4 2 300.5.b.c 4
12.b even 2 1 240.5.l.a 2
15.d odd 2 1 300.5.g.d 2
15.e even 4 2 300.5.b.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.5.g.a 2 1.a even 1 1 trivial
60.5.g.a 2 3.b odd 2 1 inner
240.5.l.a 2 4.b odd 2 1
240.5.l.a 2 12.b even 2 1
300.5.b.c 4 5.c odd 4 2
300.5.b.c 4 15.e even 4 2
300.5.g.d 2 5.b even 2 1
300.5.g.d 2 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7} - 74 \) acting on \(S_{5}^{\mathrm{new}}(60, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 81 - 12 T + T^{2} \)
$5$ \( 125 + T^{2} \)
$7$ \( ( -74 + T )^{2} \)
$11$ \( 14580 + T^{2} \)
$13$ \( ( 16 + T )^{2} \)
$17$ \( 30420 + T^{2} \)
$19$ \( ( -374 + T )^{2} \)
$23$ \( 432180 + T^{2} \)
$29$ \( 1909620 + T^{2} \)
$31$ \( ( 1426 + T )^{2} \)
$37$ \( ( -272 + T )^{2} \)
$41$ \( 605520 + T^{2} \)
$43$ \( ( 1936 + T )^{2} \)
$47$ \( 10281780 + T^{2} \)
$53$ \( 30110580 + T^{2} \)
$59$ \( 23198580 + T^{2} \)
$61$ \( ( -2234 + T )^{2} \)
$67$ \( ( 5416 + T )^{2} \)
$71$ \( 2422080 + T^{2} \)
$73$ \( ( 538 + T )^{2} \)
$79$ \( ( -9962 + T )^{2} \)
$83$ \( 21673620 + T^{2} \)
$89$ \( 27659520 + T^{2} \)
$97$ \( ( 10726 + T )^{2} \)
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