Properties

Label 60.6.j.a
Level $60$
Weight $6$
Character orbit 60.j
Analytic conductor $9.623$
Analytic rank $0$
Dimension $60$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 60 = 2^{2} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 60.j (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.62302918878\)
Analytic rank: \(0\)
Dimension: \(60\)
Relative dimension: \(30\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 60q + 180q^{6} - 492q^{8} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 60q + 180q^{6} - 492q^{8} - 568q^{10} + 792q^{12} - 244q^{13} - 2740q^{16} - 2020q^{17} + 6820q^{20} + 2212q^{22} + 15580q^{25} - 11920q^{26} - 4724q^{28} + 1368q^{30} - 25580q^{32} - 9432q^{33} + 1620q^{36} + 3764q^{37} + 9776q^{38} + 47032q^{40} + 19280q^{41} + 14220q^{42} - 6156q^{45} - 25400q^{46} - 44784q^{48} - 104816q^{50} - 61288q^{52} + 40244q^{53} + 76480q^{56} + 242100q^{58} + 88380q^{60} - 96160q^{61} + 83864q^{62} + 88420q^{65} - 63720q^{66} - 346256q^{68} - 373436q^{70} - 39852q^{72} - 72036q^{73} + 304040q^{76} + 241008q^{77} + 151416q^{78} + 534404q^{80} - 393660q^{81} + 497776q^{82} - 14276q^{85} - 719680q^{86} - 600740q^{88} - 86508q^{90} - 697064q^{92} + 1584q^{93} + 295020q^{96} + 71580q^{97} + 1425384q^{98} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −5.64888 + 0.300216i 6.36396 + 6.36396i 31.8197 3.39177i 41.9702 + 36.9256i −37.8598 34.0387i 90.7662 90.7662i −178.728 + 28.7125i 81.0000i −248.171 195.988i
7.2 −5.59359 0.843654i −6.36396 6.36396i 30.5765 + 9.43811i −51.5214 21.6921i 30.2284 + 40.9664i −47.0495 + 47.0495i −163.070 78.5889i 81.0000i 269.889 + 164.803i
7.3 −5.59263 + 0.850026i −6.36396 6.36396i 30.5549 9.50776i 52.5451 19.0791i 41.0008 + 30.1817i −61.4926 + 61.4926i −162.800 + 79.1458i 81.0000i −277.647 + 151.367i
7.4 −5.33472 + 1.88169i 6.36396 + 6.36396i 24.9185 20.0766i −7.25857 55.4285i −45.9250 21.9749i −66.6791 + 66.6791i −95.1551 + 153.992i 81.0000i 143.022 + 282.037i
7.5 −5.10793 + 2.43085i −6.36396 6.36396i 20.1820 24.8332i −21.8815 + 51.4412i 47.9765 + 17.0369i 139.056 139.056i −42.7225 + 175.906i 81.0000i −13.2765 315.949i
7.6 −4.91867 2.79405i 6.36396 + 6.36396i 16.3866 + 27.4860i −51.4784 + 21.7939i −13.5210 49.0834i −32.7439 + 32.7439i −3.80286 180.979i 81.0000i 314.098 + 36.6362i
7.7 −4.27148 3.70870i 6.36396 + 6.36396i 4.49110 + 31.6833i 43.5540 35.0435i −3.58152 50.7856i −84.5858 + 84.5858i 98.3201 151.991i 81.0000i −316.006 11.8410i
7.8 −3.70870 4.27148i −6.36396 6.36396i −4.49110 + 31.6833i 43.5540 35.0435i −3.58152 + 50.7856i 84.5858 84.5858i 151.991 98.3201i 81.0000i −311.216 56.0743i
7.9 −3.35272 + 4.55624i −6.36396 6.36396i −9.51857 30.5515i −21.8436 51.4573i 50.3323 7.65914i −30.3173 + 30.3173i 171.113 + 59.0618i 81.0000i 307.687 + 72.9970i
7.10 −3.12982 + 4.71214i 6.36396 + 6.36396i −12.4085 29.4963i −55.9013 + 0.200091i −49.9059 + 10.0698i 138.226 138.226i 177.827 + 33.8473i 81.0000i 174.018 264.041i
7.11 −2.79405 4.91867i −6.36396 6.36396i −16.3866 + 27.4860i −51.4784 + 21.7939i −13.5210 + 49.0834i 32.7439 32.7439i 180.979 + 3.80286i 81.0000i 251.030 + 192.312i
7.12 −2.54160 + 5.05374i 6.36396 + 6.36396i −19.0805 25.6892i 46.5686 + 30.9252i −48.3364 + 15.9871i −69.0576 + 69.0576i 178.321 31.1362i 81.0000i −274.647 + 156.746i
7.13 −1.55455 + 5.43906i −6.36396 6.36396i −27.1667 16.9106i −5.88437 + 55.5911i 44.5071 24.7209i −99.3719 + 99.3719i 134.210 121.473i 81.0000i −293.216 118.425i
7.14 −0.843654 5.59359i 6.36396 + 6.36396i −30.5765 + 9.43811i −51.5214 21.6921i 30.2284 40.9664i 47.0495 47.0495i 78.5889 + 163.070i 81.0000i −77.8705 + 306.490i
7.15 0.300216 5.64888i −6.36396 6.36396i −31.8197 3.39177i 41.9702 + 36.9256i −37.8598 + 34.0387i −90.7662 + 90.7662i −28.7125 + 178.728i 81.0000i 221.188 225.999i
7.16 0.446995 + 5.63917i −6.36396 6.36396i −31.6004 + 5.04136i 53.1779 + 17.2370i 33.0428 38.7321i 145.360 145.360i −42.5543 175.946i 81.0000i −73.4323 + 307.584i
7.17 0.850026 5.59263i 6.36396 + 6.36396i −30.5549 9.50776i 52.5451 19.0791i 41.0008 30.1817i 61.4926 61.4926i −79.1458 + 162.800i 81.0000i −62.0374 310.083i
7.18 1.88169 5.33472i −6.36396 6.36396i −24.9185 20.0766i −7.25857 55.4285i −45.9250 + 21.9749i 66.6791 66.6791i −153.992 + 95.1551i 81.0000i −309.354 65.5768i
7.19 2.07088 + 5.26417i 6.36396 + 6.36396i −23.4229 + 21.8029i −19.0905 + 52.5409i −20.3219 + 46.6800i 16.4587 16.4587i −163.280 78.1507i 81.0000i −316.119 + 8.31030i
7.20 2.43085 5.10793i 6.36396 + 6.36396i −20.1820 24.8332i −21.8815 + 51.4412i 47.9765 17.0369i −139.056 + 139.056i −175.906 + 42.7225i 81.0000i 209.568 + 236.815i
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.30
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.c odd 4 1 inner
20.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 60.6.j.a 60
4.b odd 2 1 inner 60.6.j.a 60
5.c odd 4 1 inner 60.6.j.a 60
20.e even 4 1 inner 60.6.j.a 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.6.j.a 60 1.a even 1 1 trivial
60.6.j.a 60 4.b odd 2 1 inner
60.6.j.a 60 5.c odd 4 1 inner
60.6.j.a 60 20.e even 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(60, [\chi])\).