Properties

Label 60.5.b.a
Level $60$
Weight $5$
Character orbit 60.b
Analytic conductor $6.202$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(6.20219778503\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Defining polynomial: \(x^{8} + 110 x^{6} + 2705 x^{4} + 17000 x^{2} + 25600\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{7}\cdot 3^{2}\cdot 5^{5} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{3} + \beta_{2} q^{5} + \beta_{6} q^{7} + ( 6 + \beta_{2} - \beta_{4} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{3} + \beta_{2} q^{5} + \beta_{6} q^{7} + ( 6 + \beta_{2} - \beta_{4} ) q^{9} + ( -\beta_{2} - \beta_{4} + \beta_{7} ) q^{11} + ( -3 \beta_{1} - \beta_{3} + 3 \beta_{6} ) q^{13} + ( -12 - 2 \beta_{1} - 2 \beta_{3} - \beta_{5} + 3 \beta_{6} + \beta_{7} ) q^{15} + ( -11 \beta_{1} + 5 \beta_{2} + 4 \beta_{3} + \beta_{5} ) q^{17} + ( 48 + \beta_{1} + 2 \beta_{2} - 5 \beta_{4} + \beta_{5} - \beta_{7} ) q^{19} + ( 27 - 2 \beta_{1} + 10 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} - \beta_{7} ) q^{21} + ( 11 \beta_{1} + 10 \beta_{2} - 3 \beta_{3} + 2 \beta_{5} ) q^{23} + ( 3 - 22 \beta_{1} - 2 \beta_{2} - 7 \beta_{3} + 5 \beta_{4} - \beta_{5} - 7 \beta_{6} + \beta_{7} ) q^{25} + ( -2 \beta_{1} - 15 \beta_{2} + 16 \beta_{3} - 3 \beta_{5} + 3 \beta_{6} ) q^{27} + ( 6 \beta_{1} - 27 \beta_{2} + 3 \beta_{4} + 6 \beta_{5} - 3 \beta_{7} ) q^{29} + ( -188 - \beta_{1} - 2 \beta_{2} + 5 \beta_{4} - \beta_{5} + \beta_{7} ) q^{31} + ( 7 \beta_{1} - 30 \beta_{2} - 13 \beta_{3} - 6 \beta_{5} - 21 \beta_{6} ) q^{33} + ( 80 \beta_{1} - 3 \beta_{2} - 25 \beta_{3} + 5 \beta_{5} ) q^{35} + ( 147 \beta_{1} + 49 \beta_{3} - 23 \beta_{6} ) q^{37} + ( -144 - 6 \beta_{1} + 33 \beta_{2} + 3 \beta_{4} - 6 \beta_{5} - 3 \beta_{7} ) q^{39} + ( 11 \beta_{1} - 48 \beta_{2} + 7 \beta_{4} + 11 \beta_{5} - 7 \beta_{7} ) q^{41} + ( -63 \beta_{1} - 21 \beta_{3} + 8 \beta_{6} ) q^{43} + ( -363 + 2 \beta_{1} + \beta_{2} - 43 \beta_{3} + 5 \beta_{4} - 9 \beta_{5} - 3 \beta_{6} - 6 \beta_{7} ) q^{45} + ( -241 \beta_{1} + 40 \beta_{2} + 83 \beta_{3} + 8 \beta_{5} ) q^{47} + ( 75 - 5 \beta_{1} - 10 \beta_{2} + 25 \beta_{4} - 5 \beta_{5} + 5 \beta_{7} ) q^{49} + ( 924 - 7 \beta_{1} + 27 \beta_{2} - 12 \beta_{4} - 7 \beta_{5} + 10 \beta_{7} ) q^{51} + ( 165 \beta_{1} + 45 \beta_{2} - 52 \beta_{3} + 9 \beta_{5} ) q^{53} + ( 932 - 243 \beta_{1} + 12 \beta_{2} - 83 \beta_{3} - 30 \beta_{4} + 6 \beta_{5} + 17 \beta_{6} - 6 \beta_{7} ) q^{55} + ( -18 \beta_{1} - 45 \beta_{2} + 129 \beta_{3} - 9 \beta_{5} + 9 \beta_{6} ) q^{57} + ( -8 \beta_{1} + 19 \beta_{2} - 21 \beta_{4} - 8 \beta_{5} + 21 \beta_{7} ) q^{59} + ( -1372 + 7 \beta_{1} + 14 \beta_{2} - 35 \beta_{4} + 7 \beta_{5} - 7 \beta_{7} ) q^{61} + ( -64 \beta_{1} + 15 \beta_{2} - 43 \beta_{3} + 3 \beta_{5} + 78 \beta_{6} ) q^{63} + ( 250 \beta_{1} - 7 \beta_{2} - 80 \beta_{3} - 5 \beta_{4} + 10 \beta_{5} + 5 \beta_{7} ) q^{65} + ( 399 \beta_{1} + 133 \beta_{3} + 104 \beta_{6} ) q^{67} + ( -1023 - 14 \beta_{1} + 21 \beta_{2} + 9 \beta_{4} - 14 \beta_{5} + 20 \beta_{7} ) q^{69} + ( -20 \beta_{1} + 86 \beta_{2} - 14 \beta_{4} - 20 \beta_{5} + 14 \beta_{7} ) q^{71} + ( -294 \beta_{1} - 98 \beta_{3} - 46 \beta_{6} ) q^{73} + ( -1764 - 19 \beta_{1} - 4 \beta_{2} - 129 \beta_{3} - 35 \beta_{4} + 23 \beta_{5} - 9 \beta_{6} + 7 \beta_{7} ) q^{75} + ( -568 \beta_{1} - 20 \beta_{2} + 188 \beta_{3} - 4 \beta_{5} ) q^{77} + ( 1884 + 3 \beta_{1} + 6 \beta_{2} - 15 \beta_{4} + 3 \beta_{5} - 3 \beta_{7} ) q^{79} + ( 4323 + 15 \beta_{1} - 16 \beta_{2} + 7 \beta_{4} + 15 \beta_{5} - 33 \beta_{7} ) q^{81} + ( 11 \beta_{1} - 110 \beta_{2} - 11 \beta_{3} - 22 \beta_{5} ) q^{83} + ( 2852 - 198 \beta_{1} - 18 \beta_{2} - 63 \beta_{3} + 45 \beta_{4} - 9 \beta_{5} + 37 \beta_{6} + 9 \beta_{7} ) q^{85} + ( 57 \beta_{1} + 180 \beta_{2} + 159 \beta_{3} + 36 \beta_{5} - 117 \beta_{6} ) q^{87} + ( -4 \beta_{1} + 62 \beta_{2} + 42 \beta_{4} - 4 \beta_{5} - 42 \beta_{7} ) q^{89} + ( -6816 - 16 \beta_{1} - 32 \beta_{2} + 80 \beta_{4} - 16 \beta_{5} + 16 \beta_{7} ) q^{91} + ( 158 \beta_{1} + 45 \beta_{2} - 129 \beta_{3} + 9 \beta_{5} - 9 \beta_{6} ) q^{93} + ( 365 \beta_{1} + 16 \beta_{2} - 140 \beta_{3} + 35 \beta_{4} - 55 \beta_{5} - 35 \beta_{7} ) q^{95} + ( 528 \beta_{1} + 176 \beta_{3} - 148 \beta_{6} ) q^{97} + ( -3084 + 84 \beta_{1} - 313 \beta_{2} - 29 \beta_{4} + 84 \beta_{5} - 39 \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 52q^{9} + O(q^{10}) \) \( 8q + 52q^{9} - 100q^{15} + 408q^{19} + 212q^{21} - 1528q^{31} - 1152q^{39} - 2900q^{45} + 480q^{49} + 7400q^{51} + 7600q^{55} - 10808q^{61} - 8300q^{69} - 14000q^{75} + 15144q^{79} + 34688q^{81} + 22600q^{85} - 54912q^{91} - 24400q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 110 x^{6} + 2705 x^{4} + 17000 x^{2} + 25600\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 49 \nu^{7} + 80 \nu^{6} + 5190 \nu^{5} + 8200 \nu^{4} + 113385 \nu^{3} + 148200 \nu^{2} + 536200 \nu + 148000 \)\()/53600\)
\(\beta_{2}\)\(=\)\((\)\( 31 \nu^{7} + 136 \nu^{6} + 3010 \nu^{5} + 15280 \nu^{4} + 45535 \nu^{3} + 379240 \nu^{2} - 120200 \nu + 1511200 \)\()/32160\)
\(\beta_{3}\)\(=\)\((\)\( 147 \nu^{7} - 240 \nu^{6} + 15570 \nu^{5} - 24600 \nu^{4} + 340155 \nu^{3} - 444600 \nu^{2} + 1608600 \nu - 444000 \)\()/53600\)
\(\beta_{4}\)\(=\)\((\)\( 133 \nu^{7} + 1096 \nu^{6} + 14470 \nu^{5} + 113680 \nu^{4} + 338005 \nu^{3} + 2318440 \nu^{2} + 1455400 \nu + 7693120 \)\()/32160\)
\(\beta_{5}\)\(=\)\((\)\( -461 \nu^{7} + 1580 \nu^{6} - 45410 \nu^{5} + 178700 \nu^{4} - 739265 \nu^{3} + 4518200 \nu^{2} + 698200 \nu + 18668000 \)\()/80400\)
\(\beta_{6}\)\(=\)\((\)\( 89 \nu^{7} + 9290 \nu^{5} + 187485 \nu^{3} + 476200 \nu \)\()/13400\)
\(\beta_{7}\)\(=\)\((\)\( -379 \nu^{7} + 616 \nu^{6} - 40690 \nu^{5} + 64480 \nu^{4} - 913315 \nu^{3} + 1348840 \nu^{2} - 3458200 \nu + 4602160 \)\()/16080\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(5 \beta_{7} + 15 \beta_{6} + 4 \beta_{5} - 5 \beta_{4} + 15 \beta_{3} - 25 \beta_{2} + 49 \beta_{1}\)\()/300\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{7} - \beta_{5} + 5 \beta_{4} + 20 \beta_{3} - 2 \beta_{2} - 61 \beta_{1} - 822\)\()/30\)
\(\nu^{3}\)\(=\)\((\)\(-25 \beta_{7} - 150 \beta_{6} - 29 \beta_{5} + 25 \beta_{4} + 170 \beta_{2} - 29 \beta_{1}\)\()/30\)
\(\nu^{4}\)\(=\)\((\)\(-95 \beta_{7} + 167 \beta_{5} - 475 \beta_{4} - 1560 \beta_{3} + 550 \beta_{2} + 4847 \beta_{1} + 49890\)\()/30\)
\(\nu^{5}\)\(=\)\((\)\(1555 \beta_{7} + 11850 \beta_{6} + 2459 \beta_{5} - 1555 \beta_{4} - 1900 \beta_{3} - 13850 \beta_{2} - 3241 \beta_{1}\)\()/30\)
\(\nu^{6}\)\(=\)\((\)\(1577 \beta_{7} - 3053 \beta_{5} + 7885 \beta_{4} + 23900 \beta_{3} - 10534 \beta_{2} - 74753 \beta_{1} - 729294\)\()/6\)
\(\nu^{7}\)\(=\)\((\)\(-22465 \beta_{7} - 184890 \beta_{6} - 39545 \beta_{5} + 22465 \beta_{4} + 38060 \beta_{3} + 220190 \beta_{2} + 74635 \beta_{1}\)\()/6\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1
8.83726i
8.83726i
4.83397i
4.83397i
1.49058i
1.49058i
2.51271i
2.51271i
0 −8.97294 0.697422i 0 9.58741 + 23.0886i 0 67.4301i 0 80.0272 + 12.5158i 0
29.2 0 −8.97294 + 0.697422i 0 9.58741 23.0886i 0 67.4301i 0 80.0272 12.5158i 0
29.3 0 −2.64318 8.60312i 0 −23.0886 + 9.58741i 0 11.6269i 0 −67.0272 + 45.4792i 0
29.4 0 −2.64318 + 8.60312i 0 −23.0886 9.58741i 0 11.6269i 0 −67.0272 45.4792i 0
29.5 0 2.64318 8.60312i 0 23.0886 9.58741i 0 11.6269i 0 −67.0272 45.4792i 0
29.6 0 2.64318 + 8.60312i 0 23.0886 + 9.58741i 0 11.6269i 0 −67.0272 + 45.4792i 0
29.7 0 8.97294 0.697422i 0 −9.58741 23.0886i 0 67.4301i 0 80.0272 12.5158i 0
29.8 0 8.97294 + 0.697422i 0 −9.58741 + 23.0886i 0 67.4301i 0 80.0272 + 12.5158i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 29.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
15.d 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.b.a 8
3.b odd 2 1 inner 60.5.b.a 8
4.b odd 2 1 240.5.c.e 8
5.b even 2 1 inner 60.5.b.a 8
5.c odd 4 2 300.5.g.h 8
12.b even 2 1 240.5.c.e 8
15.d odd 2 1 inner 60.5.b.a 8
15.e even 4 2 300.5.g.h 8
20.d odd 2 1 240.5.c.e 8
60.h even 2 1 240.5.c.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.5.b.a 8 1.a even 1 1 trivial
60.5.b.a 8 3.b odd 2 1 inner
60.5.b.a 8 5.b even 2 1 inner
60.5.b.a 8 15.d odd 2 1 inner
240.5.c.e 8 4.b odd 2 1
240.5.c.e 8 12.b even 2 1
240.5.c.e 8 20.d odd 2 1
240.5.c.e 8 60.h even 2 1
300.5.g.h 8 5.c odd 4 2
300.5.g.h 8 15.e even 4 2

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(60, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( 43046721 - 170586 T^{2} - 8334 T^{4} - 26 T^{6} + T^{8} \)
$5$ \( 152587890625 + 2750 T^{4} + T^{8} \)
$7$ \( ( 614656 + 4682 T^{2} + T^{4} )^{2} \)
$11$ \( ( 760384000 + 55600 T^{2} + T^{4} )^{2} \)
$13$ \( ( 11943936 + 42912 T^{2} + T^{4} )^{2} \)
$17$ \( ( 1236544000 - 100900 T^{2} + T^{4} )^{2} \)
$19$ \( ( -192024 - 102 T + T^{2} )^{4} \)
$23$ \( ( 21409129000 - 296350 T^{2} + T^{4} )^{2} \)
$29$ \( ( 1031180544000 + 2066400 T^{2} + T^{4} )^{2} \)
$31$ \( ( -158144 + 382 T + T^{2} )^{4} \)
$37$ \( ( 15774750401536 + 8199488 T^{2} + T^{4} )^{2} \)
$41$ \( ( 11048532544000 + 7360900 T^{2} + T^{4} )^{2} \)
$43$ \( ( 386456182336 + 1375562 T^{2} + T^{4} )^{2} \)
$47$ \( ( 137109500089000 - 23708350 T^{2} + T^{4} )^{2} \)
$53$ \( ( 21069103104000 - 14984100 T^{2} + T^{4} )^{2} \)
$59$ \( ( 40800768064000 + 22135600 T^{2} + T^{4} )^{2} \)
$61$ \( ( -7711424 + 2702 T + T^{2} )^{4} \)
$67$ \( ( 2717572765558336 + 106879562 T^{2} + T^{4} )^{2} \)
$71$ \( ( 114709561344000 + 25257600 T^{2} + T^{4} )^{2} \)
$73$ \( ( 226681208836096 + 38532128 T^{2} + T^{4} )^{2} \)
$79$ \( ( 1831824 - 3786 T + T^{2} )^{4} \)
$83$ \( ( 63525673849000 - 29905150 T^{2} + T^{4} )^{2} \)
$89$ \( ( 2771452493824000 + 105462400 T^{2} + T^{4} )^{2} \)
$97$ \( ( 6227548184313856 + 169065632 T^{2} + T^{4} )^{2} \)
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