Properties

Label 76.8.d.a
Level $76$
Weight $8$
Character orbit 76.d
Analytic conductor $23.741$
Analytic rank $0$
Dimension $68$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 76 = 2^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 76.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(23.7412619368\)
Analytic rank: \(0\)
Dimension: \(68\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 68q - 54q^{4} - 4q^{5} - 150q^{6} + 48888q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 68q - 54q^{4} - 4q^{5} - 150q^{6} + 48888q^{9} + 13050q^{16} + 23968q^{17} - 79660q^{20} - 232154q^{24} + 937496q^{25} - 391242q^{26} - 37666q^{28} - 468496q^{30} - 1128652q^{36} - 530466q^{38} + 651806q^{42} + 1433776q^{44} - 321252q^{45} - 4014380q^{49} + 1697586q^{54} + 137420q^{57} + 2227894q^{58} - 3042164q^{61} + 2261524q^{62} + 9825150q^{64} - 749476q^{66} - 11176962q^{68} + 598424q^{73} - 6195964q^{74} - 1714536q^{76} - 2556764q^{77} + 16623216q^{80} + 43726748q^{81} - 20460388q^{82} + 27222844q^{85} - 61733582q^{92} + 17778200q^{93} - 36370886q^{96} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
75.1 −11.3050 0.442790i 16.6522 127.608 + 10.0115i 341.116 −188.254 7.37345i 1382.96i −1438.18 169.684i −1909.70 −3856.33 151.043i
75.2 −11.3050 + 0.442790i 16.6522 127.608 10.0115i 341.116 −188.254 + 7.37345i 1382.96i −1438.18 + 169.684i −1909.70 −3856.33 + 151.043i
75.3 −11.1927 1.65015i −65.7104 122.554 + 36.9394i −95.9690 735.478 + 108.432i 851.669i −1310.76 615.686i 2130.86 1074.15 + 158.364i
75.4 −11.1927 + 1.65015i −65.7104 122.554 36.9394i −95.9690 735.478 108.432i 851.669i −1310.76 + 615.686i 2130.86 1074.15 158.364i
75.5 −10.9753 2.74630i −8.30507 112.916 + 60.2831i −360.155 91.1509 + 22.8082i 965.147i −1073.73 971.727i −2118.03 3952.82 + 989.094i
75.6 −10.9753 + 2.74630i −8.30507 112.916 60.2831i −360.155 91.1509 22.8082i 965.147i −1073.73 + 971.727i −2118.03 3952.82 989.094i
75.7 −10.7294 3.58884i 80.7260 102.240 + 77.0123i 174.727 −866.143 289.713i 433.835i −820.594 1193.22i 4329.69 −1874.71 627.067i
75.8 −10.7294 + 3.58884i 80.7260 102.240 77.0123i 174.727 −866.143 + 289.713i 433.835i −820.594 + 1193.22i 4329.69 −1874.71 + 627.067i
75.9 −10.6582 3.79499i 60.8414 99.1961 + 80.8959i −369.508 −648.462 230.893i 900.474i −750.256 1238.66i 1514.68 3938.30 + 1402.28i
75.10 −10.6582 + 3.79499i 60.8414 99.1961 80.8959i −369.508 −648.462 + 230.893i 900.474i −750.256 + 1238.66i 1514.68 3938.30 1402.28i
75.11 −9.80435 5.64577i −65.3605 64.2506 + 110.706i 399.628 640.817 + 369.010i 321.615i −4.91377 1448.15i 2084.99 −3918.10 2256.21i
75.12 −9.80435 + 5.64577i −65.3605 64.2506 110.706i 399.628 640.817 369.010i 321.615i −4.91377 + 1448.15i 2084.99 −3918.10 + 2256.21i
75.13 −9.59747 5.99070i −1.36022 56.2230 + 114.991i 59.9018 13.0547 + 8.14867i 254.899i 149.279 1440.44i −2185.15 −574.906 358.854i
75.14 −9.59747 + 5.99070i −1.36022 56.2230 114.991i 59.9018 13.0547 8.14867i 254.899i 149.279 + 1440.44i −2185.15 −574.906 + 358.854i
75.15 −8.06300 7.93650i 49.8474 2.02391 + 127.984i 33.4044 −401.919 395.614i 865.446i 999.426 1048.00i 297.759 −269.340 265.114i
75.16 −8.06300 + 7.93650i 49.8474 2.02391 127.984i 33.4044 −401.919 + 395.614i 865.446i 999.426 + 1048.00i 297.759 −269.340 + 265.114i
75.17 −7.73771 8.25396i −29.6173 −8.25557 + 127.733i −224.326 229.170 + 244.460i 1196.17i 1118.19 920.224i −1309.82 1735.77 + 1851.58i
75.18 −7.73771 + 8.25396i −29.6173 −8.25557 127.733i −224.326 229.170 244.460i 1196.17i 1118.19 + 920.224i −1309.82 1735.77 1851.58i
75.19 −7.35548 8.59633i −83.0232 −19.7938 + 126.460i −447.252 610.676 + 713.695i 106.200i 1232.69 760.022i 4705.86 3289.75 + 3844.72i
75.20 −7.35548 + 8.59633i −83.0232 −19.7938 126.460i −447.252 610.676 713.695i 106.200i 1232.69 + 760.022i 4705.86 3289.75 3844.72i
See all 68 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 75.68
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
19.b odd 2 1 inner
76.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 76.8.d.a 68
4.b odd 2 1 inner 76.8.d.a 68
19.b odd 2 1 inner 76.8.d.a 68
76.d even 2 1 inner 76.8.d.a 68
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.8.d.a 68 1.a even 1 1 trivial
76.8.d.a 68 4.b odd 2 1 inner
76.8.d.a 68 19.b odd 2 1 inner
76.8.d.a 68 76.d even 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(76, [\chi])\).