Properties

Label 60.6.h.a
Level $60$
Weight $6$
Character orbit 60.h
Analytic conductor $9.623$
Analytic rank $0$
Dimension $4$
CM discriminant -20
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 60 = 2^{2} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 60.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.62302918878\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-5})\)
Defining polynomial: \(x^{4} - 4 x^{2} + 9\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 5^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + 4 \beta_{1} q^{2} + ( 10 \beta_{1} + \beta_{2} ) q^{3} -32 q^{4} + 5 \beta_{3} q^{5} + ( -76 + 4 \beta_{3} ) q^{6} + ( -\beta_{1} - 2 \beta_{2} ) q^{7} -128 \beta_{1} q^{8} + ( -118 + 19 \beta_{3} ) q^{9} +O(q^{10})\) \( q + 4 \beta_{1} q^{2} + ( 10 \beta_{1} + \beta_{2} ) q^{3} -32 q^{4} + 5 \beta_{3} q^{5} + ( -76 + 4 \beta_{3} ) q^{6} + ( -\beta_{1} - 2 \beta_{2} ) q^{7} -128 \beta_{1} q^{8} + ( -118 + 19 \beta_{3} ) q^{9} + ( -20 \beta_{1} - 40 \beta_{2} ) q^{10} + ( -320 \beta_{1} - 32 \beta_{2} ) q^{12} -8 \beta_{3} q^{14} + ( 265 \beta_{1} - 95 \beta_{2} ) q^{15} + 1024 q^{16} + ( -548 \beta_{1} - 152 \beta_{2} ) q^{18} -160 \beta_{3} q^{20} + ( -125 - 19 \beta_{3} ) q^{21} -2419 \beta_{1} q^{23} + ( 2432 - 128 \beta_{3} ) q^{24} -3125 q^{25} + ( -173 \beta_{1} - 479 \beta_{2} ) q^{27} + ( 32 \beta_{1} + 64 \beta_{2} ) q^{28} + 796 \beta_{3} q^{29} + ( -2500 - 380 \beta_{3} ) q^{30} + 4096 \beta_{1} q^{32} -625 \beta_{1} q^{35} + ( 3776 - 608 \beta_{3} ) q^{36} + ( 640 \beta_{1} + 1280 \beta_{2} ) q^{40} + 1882 \beta_{3} q^{41} + ( -424 \beta_{1} + 152 \beta_{2} ) q^{42} + ( 1439 \beta_{1} + 2878 \beta_{2} ) q^{43} + ( -11875 - 590 \beta_{3} ) q^{45} + 19352 q^{46} + 16667 \beta_{1} q^{47} + ( 10240 \beta_{1} + 1024 \beta_{2} ) q^{48} -16557 q^{49} -12500 \beta_{1} q^{50} + ( -532 - 1916 \beta_{3} ) q^{54} + 256 \beta_{3} q^{56} + ( -3184 \beta_{1} - 6368 \beta_{2} ) q^{58} + ( -8480 \beta_{1} + 3040 \beta_{2} ) q^{60} + 25448 q^{61} + ( -2257 \beta_{1} + 236 \beta_{2} ) q^{63} -32768 q^{64} + ( -1195 \beta_{1} - 2390 \beta_{2} ) q^{67} + ( 45961 - 2419 \beta_{3} ) q^{69} + 5000 q^{70} + ( 17536 \beta_{1} + 4864 \beta_{2} ) q^{72} + ( -31250 \beta_{1} - 3125 \beta_{2} ) q^{75} + 5120 \beta_{3} q^{80} + ( -31201 - 4484 \beta_{3} ) q^{81} + ( -7528 \beta_{1} - 15056 \beta_{2} ) q^{82} -81631 \beta_{1} q^{83} + ( 4000 + 608 \beta_{3} ) q^{84} + 11512 \beta_{3} q^{86} + ( 42188 \beta_{1} - 15124 \beta_{2} ) q^{87} -632 \beta_{3} q^{89} + ( -45140 \beta_{1} + 4720 \beta_{2} ) q^{90} + 77408 \beta_{1} q^{92} -133336 q^{94} + ( -77824 + 4096 \beta_{3} ) q^{96} -66228 \beta_{1} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 128q^{4} - 304q^{6} - 472q^{9} + O(q^{10}) \) \( 4q - 128q^{4} - 304q^{6} - 472q^{9} + 4096q^{16} - 500q^{21} + 9728q^{24} - 12500q^{25} - 10000q^{30} + 15104q^{36} - 47500q^{45} + 77408q^{46} - 66228q^{49} - 2128q^{54} + 101792q^{61} - 131072q^{64} + 183844q^{69} + 20000q^{70} - 124804q^{81} + 16000q^{84} - 533344q^{94} - 311296q^{96} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 4 x^{2} + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} - \nu \)\()/3\)
\(\beta_{2}\)\(=\)\( -\nu^{3} + 6 \nu \)
\(\beta_{3}\)\(=\)\( 5 \nu^{2} - 10 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + 3 \beta_{1}\)\()/5\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + 10\)\()/5\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{2} + 18 \beta_{1}\)\()/5\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
59.1
−1.58114 0.707107i
1.58114 0.707107i
−1.58114 + 0.707107i
1.58114 + 0.707107i
5.65685i −7.90569 13.4350i −32.0000 55.9017i −76.0000 + 44.7214i 15.8114 181.019i −118.000 + 212.426i 316.228
59.2 5.65685i 7.90569 13.4350i −32.0000 55.9017i −76.0000 44.7214i −15.8114 181.019i −118.000 212.426i −316.228
59.3 5.65685i −7.90569 + 13.4350i −32.0000 55.9017i −76.0000 44.7214i 15.8114 181.019i −118.000 212.426i 316.228
59.4 5.65685i 7.90569 + 13.4350i −32.0000 55.9017i −76.0000 + 44.7214i −15.8114 181.019i −118.000 + 212.426i −316.228
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)
3.b odd 2 1 inner
4.b odd 2 1 inner
5.b even 2 1 inner
12.b even 2 1 inner
15.d odd 2 1 inner
60.h even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 60.6.h.a 4
3.b odd 2 1 inner 60.6.h.a 4
4.b odd 2 1 inner 60.6.h.a 4
5.b even 2 1 inner 60.6.h.a 4
12.b even 2 1 inner 60.6.h.a 4
15.d odd 2 1 inner 60.6.h.a 4
20.d odd 2 1 CM 60.6.h.a 4
60.h even 2 1 inner 60.6.h.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.6.h.a 4 1.a even 1 1 trivial
60.6.h.a 4 3.b odd 2 1 inner
60.6.h.a 4 4.b odd 2 1 inner
60.6.h.a 4 5.b even 2 1 inner
60.6.h.a 4 12.b even 2 1 inner
60.6.h.a 4 15.d odd 2 1 inner
60.6.h.a 4 20.d odd 2 1 CM
60.6.h.a 4 60.h even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} - 250 \) acting on \(S_{6}^{\mathrm{new}}(60, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 32 + T^{2} )^{2} \)
$3$ \( 59049 + 236 T^{2} + T^{4} \)
$5$ \( ( 3125 + T^{2} )^{2} \)
$7$ \( ( -250 + T^{2} )^{2} \)
$11$ \( T^{4} \)
$13$ \( T^{4} \)
$17$ \( T^{4} \)
$19$ \( T^{4} \)
$23$ \( ( 11703122 + T^{2} )^{2} \)
$29$ \( ( 79202000 + T^{2} )^{2} \)
$31$ \( T^{4} \)
$37$ \( T^{4} \)
$41$ \( ( 442740500 + T^{2} )^{2} \)
$43$ \( ( -517680250 + T^{2} )^{2} \)
$47$ \( ( 555577778 + T^{2} )^{2} \)
$53$ \( T^{4} \)
$59$ \( T^{4} \)
$61$ \( ( -25448 + T )^{4} \)
$67$ \( ( -357006250 + T^{2} )^{2} \)
$71$ \( T^{4} \)
$73$ \( T^{4} \)
$79$ \( T^{4} \)
$83$ \( ( 13327240322 + T^{2} )^{2} \)
$89$ \( ( 49928000 + T^{2} )^{2} \)
$97$ \( T^{4} \)
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