Properties

Label 60.7.l.a
Level $60$
Weight $7$
Character orbit 60.l
Analytic conductor $13.803$
Analytic rank $0$
Dimension $136$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 60 = 2^{2} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 60.l (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(13.8032450172\)
Analytic rank: \(0\)
Dimension: \(136\)
Relative dimension: \(68\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 136q - 4q^{6} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 136q - 4q^{6} - 2092q^{10} - 1460q^{12} - 8q^{13} - 932q^{16} - 12744q^{18} - 16056q^{21} + 25444q^{22} - 8q^{25} - 52244q^{28} + 82868q^{30} - 2920q^{33} - 69524q^{36} - 113000q^{37} + 99944q^{40} + 19436q^{42} - 62504q^{45} + 145936q^{46} - 82364q^{48} + 219504q^{52} - 286152q^{57} - 351260q^{58} + 466756q^{60} + 652976q^{61} - 61800q^{66} + 243868q^{70} - 145560q^{72} - 316856q^{73} + 269024q^{76} + 613272q^{78} + 1472232q^{81} - 1652880q^{82} - 1466984q^{85} - 1084100q^{88} + 448368q^{90} - 2869552q^{93} - 278860q^{96} + 2135272q^{97} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
23.1 −7.99914 + 0.117326i −11.4327 24.4601i 63.9725 1.87701i 124.485 11.3316i 94.3213 + 194.318i 152.267 + 152.267i −511.504 + 22.5201i −467.588 + 559.287i −994.446 + 105.248i
23.2 −7.95256 + 0.869892i 26.9945 + 0.547129i 62.4866 13.8357i −116.176 46.1326i −215.151 + 19.1312i 371.962 + 371.962i −484.893 + 164.386i 728.401 + 29.5389i 964.025 + 265.812i
23.3 −7.93034 + 1.05340i −26.8514 + 2.82881i 61.7807 16.7077i −46.5899 115.993i 209.961 50.7188i 112.704 + 112.704i −472.342 + 197.578i 712.996 151.915i 491.661 + 870.787i
23.4 −7.90644 1.21994i 25.6796 + 8.34012i 61.0235 + 19.2908i 88.2710 88.5055i −192.860 97.2682i −452.639 452.639i −458.945 226.966i 589.885 + 428.342i −805.881 + 592.078i
23.5 −7.81252 1.72179i 14.1892 22.9710i 58.0709 + 26.9030i −24.8314 + 122.509i −150.404 + 155.031i −49.7637 49.7637i −407.358 310.166i −326.336 651.879i 404.930 914.348i
23.6 −7.74745 + 1.99426i −26.4363 + 5.48844i 56.0458 30.9009i 29.1126 + 121.563i 193.868 95.2423i −407.374 407.374i −372.588 + 351.173i 668.754 290.188i −467.976 883.741i
23.7 −7.71467 2.11750i 0.579762 + 26.9938i 55.0324 + 32.6717i −123.898 + 16.5611i 52.6868 209.476i −163.235 163.235i −355.374 368.583i −728.328 + 31.2999i 990.901 + 134.591i
23.8 −7.60301 2.48882i −13.5986 + 23.3255i 51.6115 + 37.8451i 120.360 + 33.7399i 161.443 143.500i 184.076 + 184.076i −298.214 416.188i −359.157 634.387i −831.129 556.080i
23.9 −7.21208 + 3.46207i 16.1580 + 21.6314i 40.0282 49.9374i 17.7152 + 123.738i −191.422 100.068i 0.599925 + 0.599925i −115.800 + 498.733i −206.838 + 699.041i −556.154 831.079i
23.10 −7.19331 + 3.50090i −0.790775 + 26.9884i 39.4874 50.3661i 69.0780 104.179i −88.7955 196.904i 138.578 + 138.578i −107.718 + 500.540i −727.749 42.6835i −132.179 + 991.226i
23.11 −6.97975 + 3.90937i 11.0251 24.6464i 33.4337 54.5728i −92.0582 84.5593i 19.3996 + 215.127i −403.029 403.029i −20.0136 + 511.609i −485.894 543.459i 973.117 + 230.313i
23.12 −6.94862 3.96443i −11.8150 24.2777i 32.5666 + 55.0946i −69.7748 103.713i −14.1489 + 215.536i −171.027 171.027i −7.87369 511.939i −449.810 + 573.683i 73.6732 + 997.282i
23.13 −6.64350 4.45690i −26.9904 0.719136i 24.2722 + 59.2188i −98.2829 + 77.2365i 176.106 + 125.071i 282.830 + 282.830i 102.680 501.598i 727.966 + 38.8196i 997.177 75.0842i
23.14 −6.59925 + 4.52216i −14.2278 22.9471i 23.1001 59.6857i −92.2676 + 84.3308i 197.663 + 87.0928i 234.474 + 234.474i 117.465 + 498.343i −324.137 + 652.975i 227.539 973.769i
23.15 −6.29771 + 4.93344i 22.6035 14.7676i 15.3224 62.1388i 124.886 + 5.34166i −69.4955 + 204.515i 79.3716 + 79.3716i 210.062 + 466.924i 292.837 667.598i −812.848 + 582.476i
23.16 −6.25131 4.99210i 21.8589 + 15.8490i 14.1578 + 62.4144i 88.9335 + 87.8398i −57.5269 208.199i 252.976 + 252.976i 223.074 460.849i 226.620 + 692.881i −117.447 993.079i
23.17 −6.11516 5.15798i 20.2023 17.9128i 10.7904 + 63.0838i 49.4715 114.794i −215.934 + 5.33675i 261.181 + 261.181i 259.400 441.424i 87.2628 723.758i −894.630 + 446.808i
23.18 −5.15798 6.11516i −20.2023 + 17.9128i −10.7904 + 63.0838i 49.4715 114.794i 213.743 + 31.1461i −261.181 261.181i 441.424 259.400i 87.2628 723.758i −957.155 + 289.577i
23.19 −4.99210 6.25131i −21.8589 15.8490i −14.1578 + 62.4144i 88.9335 + 87.8398i 10.0448 + 215.766i −252.976 252.976i 460.849 223.074i 226.620 + 692.881i 105.149 994.457i
23.20 −4.93344 + 6.29771i −14.7676 + 22.6035i −15.3224 62.1388i −124.886 5.34166i −69.4955 204.515i −79.3716 79.3716i 466.924 + 210.062i −292.837 667.598i 649.757 760.142i
See next 80 embeddings (of 136 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 47.68
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
5.c odd 4 1 inner
12.b even 2 1 inner
15.e even 4 1 inner
20.e even 4 1 inner
60.l odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 60.7.l.a 136
3.b odd 2 1 inner 60.7.l.a 136
4.b odd 2 1 inner 60.7.l.a 136
5.c odd 4 1 inner 60.7.l.a 136
12.b even 2 1 inner 60.7.l.a 136
15.e even 4 1 inner 60.7.l.a 136
20.e even 4 1 inner 60.7.l.a 136
60.l odd 4 1 inner 60.7.l.a 136
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.7.l.a 136 1.a even 1 1 trivial
60.7.l.a 136 3.b odd 2 1 inner
60.7.l.a 136 4.b odd 2 1 inner
60.7.l.a 136 5.c odd 4 1 inner
60.7.l.a 136 12.b even 2 1 inner
60.7.l.a 136 15.e even 4 1 inner
60.7.l.a 136 20.e even 4 1 inner
60.7.l.a 136 60.l odd 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(60, [\chi])\).