Properties

Label 60.6.e.a
Level $60$
Weight $6$
Character orbit 60.e
Analytic conductor $9.623$
Analytic rank $0$
Dimension $40$
CM no
Inner twists $4$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(9.62302918878\)
Analytic rank: \(0\)
Dimension: \(40\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40q - 10q^{4} - 114q^{6} + 44q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 40q - 10q^{4} - 114q^{6} + 44q^{9} + 50q^{10} + 1876q^{12} - 232q^{13} - 2734q^{16} - 3536q^{18} + 820q^{21} + 13684q^{22} + 9550q^{24} - 25000q^{25} - 23188q^{28} - 5800q^{30} - 9424q^{33} + 38972q^{34} + 53754q^{36} + 2152q^{37} - 10850q^{40} - 62868q^{42} + 5900q^{45} + 78748q^{46} + 100956q^{48} - 52144q^{49} - 82528q^{52} - 120862q^{54} - 44424q^{57} + 97476q^{58} + 3850q^{60} + 2072q^{61} - 91234q^{64} - 45784q^{66} + 52764q^{69} - 3300q^{70} - 10344q^{72} + 264960q^{73} + 73356q^{76} + 176440q^{78} - 165120q^{81} - 170920q^{82} - 250544q^{84} - 132400q^{85} + 359116q^{88} + 35550q^{90} + 510368q^{93} - 623100q^{94} - 379254q^{96} + 453648q^{97} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1 −5.57370 0.966346i −14.2123 + 6.40394i 30.1324 + 10.7723i 25.0000i 85.4036 21.9597i 68.6753i −157.539 89.1596i 160.979 182.030i −24.1587 + 139.343i
11.2 −5.57370 + 0.966346i −14.2123 6.40394i 30.1324 10.7723i 25.0000i 85.4036 + 21.9597i 68.6753i −157.539 + 89.1596i 160.979 + 182.030i −24.1587 139.343i
11.3 −5.54388 1.12492i 15.2205 3.36711i 29.4691 + 12.4728i 25.0000i −88.1681 + 1.54508i 209.419i −149.342 102.298i 220.325 102.498i 28.1229 138.597i
11.4 −5.54388 + 1.12492i 15.2205 + 3.36711i 29.4691 12.4728i 25.0000i −88.1681 1.54508i 209.419i −149.342 + 102.298i 220.325 + 102.498i 28.1229 + 138.597i
11.5 −5.05574 2.53762i −1.23746 15.5393i 19.1209 + 25.6591i 25.0000i −33.1765 + 81.7026i 220.312i −31.5574 178.247i −239.937 + 38.4585i 63.4406 126.393i
11.6 −5.05574 + 2.53762i −1.23746 + 15.5393i 19.1209 25.6591i 25.0000i −33.1765 81.7026i 220.312i −31.5574 + 178.247i −239.937 38.4585i 63.4406 + 126.393i
11.7 −4.53291 3.38418i 12.7254 9.00363i 9.09463 + 30.6804i 25.0000i −88.1529 2.25224i 20.4870i 62.6029 169.850i 80.8694 229.149i −84.6045 + 113.323i
11.8 −4.53291 + 3.38418i 12.7254 + 9.00363i 9.09463 30.6804i 25.0000i −88.1529 + 2.25224i 20.4870i 62.6029 + 169.850i 80.8694 + 229.149i −84.6045 113.323i
11.9 −4.48243 3.45077i −10.3661 11.6423i 8.18436 + 30.9357i 25.0000i 6.29061 + 87.9570i 180.345i 70.0661 166.909i −28.0862 + 241.371i −86.2693 + 112.061i
11.10 −4.48243 + 3.45077i −10.3661 + 11.6423i 8.18436 30.9357i 25.0000i 6.29061 87.9570i 180.345i 70.0661 + 166.909i −28.0862 241.371i −86.2693 112.061i
11.11 −3.72823 4.25445i −11.4665 + 10.5602i −4.20063 + 31.7231i 25.0000i 87.6778 + 9.41281i 94.6704i 150.625 100.400i 19.9629 242.179i 106.361 93.2057i
11.12 −3.72823 + 4.25445i −11.4665 10.5602i −4.20063 31.7231i 25.0000i 87.6778 9.41281i 94.6704i 150.625 + 100.400i 19.9629 + 242.179i 106.361 + 93.2057i
11.13 −3.19778 4.66628i 8.70197 + 12.9335i −11.5484 + 29.8435i 25.0000i 32.5244 81.9644i 40.1977i 176.187 41.5448i −91.5514 + 225.094i −116.657 + 79.9445i
11.14 −3.19778 + 4.66628i 8.70197 12.9335i −11.5484 29.8435i 25.0000i 32.5244 + 81.9644i 40.1977i 176.187 + 41.5448i −91.5514 225.094i −116.657 79.9445i
11.15 −2.24299 5.19317i 15.5136 + 1.52631i −21.9379 + 23.2965i 25.0000i −26.8704 83.9880i 85.2589i 170.189 + 61.6735i 238.341 + 47.3571i 129.829 56.0749i
11.16 −2.24299 + 5.19317i 15.5136 1.52631i −21.9379 23.2965i 25.0000i −26.8704 + 83.9880i 85.2589i 170.189 61.6735i 238.341 47.3571i 129.829 + 56.0749i
11.17 −1.26203 5.51428i −1.23728 15.5393i −28.8145 + 13.9184i 25.0000i −84.1264 + 26.4338i 173.531i 113.115 + 141.326i −239.938 + 38.4528i 137.857 31.5508i
11.18 −1.26203 + 5.51428i −1.23728 + 15.5393i −28.8145 13.9184i 25.0000i −84.1264 26.4338i 173.531i 113.115 141.326i −239.938 38.4528i 137.857 + 31.5508i
11.19 −0.00749860 5.65685i 8.15586 13.2847i −31.9999 + 0.0848369i 25.0000i −75.2104 46.0368i 55.6023i 0.719864 + 181.018i −109.964 216.695i −141.421 + 0.187465i
11.20 −0.00749860 + 5.65685i 8.15586 + 13.2847i −31.9999 0.0848369i 25.0000i −75.2104 + 46.0368i 55.6023i 0.719864 181.018i −109.964 + 216.695i −141.421 0.187465i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.40
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
12.b 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.e.a 40
3.b odd 2 1 inner 60.6.e.a 40
4.b odd 2 1 inner 60.6.e.a 40
12.b even 2 1 inner 60.6.e.a 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.6.e.a 40 1.a even 1 1 trivial
60.6.e.a 40 3.b odd 2 1 inner
60.6.e.a 40 4.b odd 2 1 inner
60.6.e.a 40 12.b even 2 1 inner

Hecke kernels

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