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e.g. 3,7,19 e.g. F3=1-T,F5=1+T+5T^2

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$L(s) = \prod_p F_p(p^{-s})^{-1}$
Label Origin \(F_{ 2 }(T)\) \(F_{ 3 }(T)\) \(F_{ 5 }(T)\) \(F_{ 7 }(T)\)
4-2900-1.1-c1e2-0-1 Elliptic curve 2.2.5.1-116.1-b Elliptic curve 2.2.5.1-116.2-b Hilbert modular form 2.2.5.1-116.1-b Hilbert modular form 2.2.5.1-116.2-b $( 1 - T )( 1 + T )$ $1 + 4 T^{2} + 9 T^{4}$ $1 + T + 5 T^{2}$ $1 - T^{2} + 49 T^{4}$
4-3728-1.1-c1e2-0-0 Bianchi modular form 2.0.4.1-233.1-a Bianchi modular form 2.0.4.1-233.2-a Elliptic curve 2.0.4.1-233.1-a Elliptic curve 2.0.4.1-233.2-a $1 + 2 T + 2 T^{2}$ $1 + 4 T^{2} + 9 T^{4}$ $( 1 + 2 T + 5 T^{2} )( 1 + 3 T + 5 T^{2} )$ $1 - T^{2} + 49 T^{4}$
4-10532-1.1-c1e2-0-0 Genus 2 curve 10532.a $( 1 - T )( 1 + T )$ $( 1 - T + 3 T^{2} )( 1 + 2 T + 3 T^{2} )$ $1 + 4 T + 12 T^{2} + 20 T^{3} + 25 T^{4}$ $1 - T^{2} + 49 T^{4}$
4-10658-1.1-c1e2-0-0 Elliptic curve 2.2.73.1-2.1-a Elliptic curve 2.2.73.1-2.2-a Hilbert modular form 2.2.73.1-2.1-a Hilbert modular form 2.2.73.1-2.2-a $( 1 + T )( 1 - 2 T + 2 T^{2} )$ $( 1 - 3 T + 3 T^{2} )( 1 + 2 T + 3 T^{2} )$ $1 + 2 T^{2} + 25 T^{4}$ $1 - T^{2} + 49 T^{4}$
4-104e2-1.1-c1e2-0-8 Elliptic curve 2.2.13.1-64.1-a Elliptic curve 2.2.13.1-64.1-b Hilbert modular form 2.2.13.1-64.1-a Hilbert modular form 2.2.13.1-64.1-b $1$ $( 1 - 3 T + 3 T^{2} )( 1 + T + 3 T^{2} )$ $1 + 7 T^{2} + 25 T^{4}$ $1 - T^{2} + 49 T^{4}$
4-11383-1.1-c1e2-0-0 Genus 2 curve 11383.a $1 + 2 T + 3 T^{2} + 4 T^{3} + 4 T^{4}$ $( 1 + T + 3 T^{2} )( 1 + 3 T + 3 T^{2} )$ $( 1 - T + 5 T^{2} )( 1 + 3 T + 5 T^{2} )$ $1 - T^{2} + 49 T^{4}$
4-110e2-1.1-c1e2-0-7 Elliptic curve 2.2.5.1-484.2-b Elliptic curve 2.2.5.1-484.3-b Hilbert modular form 2.2.5.1-484.2-b Hilbert modular form 2.2.5.1-484.3-b $1 + T^{2}$ $( 1 - T + 3 T^{2} )( 1 + T + 3 T^{2} )$ $1 + 3 T + 5 T^{2}$ $1 - T^{2} + 49 T^{4}$
4-13725-1.1-c1e2-0-2 Genus 2 curve 13725.a $1 + 2 T + 3 T^{2} + 4 T^{3} + 4 T^{4}$ $( 1 - T )( 1 + T )$ $1 + T + T^{2}$ $1 - T^{2} + 49 T^{4}$
4-15440-1.1-c1e2-0-1 Bianchi modular form 2.0.4.1-965.1-a Bianchi modular form 2.0.4.1-965.4-a Elliptic curve 2.0.4.1-965.1-a Elliptic curve 2.0.4.1-965.4-a $1 + 2 T + 2 T^{2}$ $1 + 3 T^{2} + 9 T^{4}$ $( 1 - T )( 1 + T + 5 T^{2} )$ $1 - T^{2} + 49 T^{4}$
4-15548-1.1-c1e2-0-1 Elliptic curve 2.2.13.1-92.1-b Elliptic curve 2.2.13.1-92.2-b Hilbert modular form 2.2.13.1-92.1-b Hilbert modular form 2.2.13.1-92.2-b $( 1 - T )( 1 + T )$ $( 1 + T + 3 T^{2} )( 1 + 3 T + 3 T^{2} )$ $1 - 3 T^{2} + 25 T^{4}$ $1 - T^{2} + 49 T^{4}$
4-16092-1.1-c1e2-0-0 Genus 2 curve 16092.a $( 1 - T )( 1 + T )$ $1 + 3 T + 3 T^{2}$ $1 + T + 3 T^{2} + 5 T^{3} + 25 T^{4}$ $1 - T^{2} + 49 T^{4}$
4-17149-1.1-c1e2-0-0 Genus 2 curve 17149.b $1 + T + 3 T^{2} + 2 T^{3} + 4 T^{4}$ $1 + T - T^{2} + 3 T^{3} + 9 T^{4}$ $1 + 2 T + 6 T^{2} + 10 T^{3} + 25 T^{4}$ $1 - T^{2} + 49 T^{4}$
4-18968-1.1-c1e2-0-0 Genus 2 curve 18968.a $1 - T$ $( 1 - T + 3 T^{2} )( 1 + 3 T + 3 T^{2} )$ $1 + 3 T + 4 T^{2} + 15 T^{3} + 25 T^{4}$ $1 - T^{2} + 49 T^{4}$
4-19375-1.1-c1e2-0-4 Elliptic curve 2.2.5.1-775.1-b Elliptic curve 2.2.5.1-775.2-b Hilbert modular form 2.2.5.1-775.1-b Hilbert modular form 2.2.5.1-775.2-b $1 + 2 T^{2} + 4 T^{4}$ $1 + 4 T^{2} + 9 T^{4}$ $1$ $1 - T^{2} + 49 T^{4}$
4-20448-1.1-c1e2-0-0 Elliptic curve 2.2.12.1-142.1-b Elliptic curve 2.2.12.1-142.2-b Hilbert modular form 2.2.12.1-142.1-b Hilbert modular form 2.2.12.1-142.2-b $1 - T$ $1 + 3 T + 3 T^{2}$ $1 + 2 T^{2} + 25 T^{4}$ $1 - T^{2} + 49 T^{4}$
4-20448-1.1-c1e2-0-3 Elliptic curve 2.2.12.1-142.1-c Elliptic curve 2.2.12.1-142.2-c Hilbert modular form 2.2.12.1-142.1-c Hilbert modular form 2.2.12.1-142.2-c $1 + T$ $1 - 3 T + 3 T^{2}$ $1 + 2 T^{2} + 25 T^{4}$ $1 - T^{2} + 49 T^{4}$
4-21632-1.1-c1e2-0-4 Bianchi modular form 2.0.4.1-1352.1-a Bianchi modular form 2.0.4.1-1352.3-a Elliptic curve 2.0.4.1-1352.1-a Elliptic curve 2.0.4.1-1352.3-a $1$ $1 + T^{2} + 9 T^{4}$ $( 1 + T + 5 T^{2} )( 1 + 3 T + 5 T^{2} )$ $1 - T^{2} + 49 T^{4}$
4-22375-1.1-c1e2-0-2 Elliptic curve 2.2.5.1-895.1-c Elliptic curve 2.2.5.1-895.2-c Hilbert modular form 2.2.5.1-895.1-c Hilbert modular form 2.2.5.1-895.2-c $1 + 2 T^{2} + 4 T^{4}$ $1 + 9 T^{4}$ $1 - T$ $1 - T^{2} + 49 T^{4}$
4-22625-1.1-c1e2-0-2 Elliptic curve 2.2.5.1-905.1-a Elliptic curve 2.2.5.1-905.2-a Hilbert modular form 2.2.5.1-905.1-a Hilbert modular form 2.2.5.1-905.2-a $1 + 2 T^{2} + 4 T^{4}$ $1 + 4 T^{2} + 9 T^{4}$ $1 - T$ $1 - T^{2} + 49 T^{4}$
4-24111-1.1-c1e2-0-0 Genus 2 curve 24111.a $1 + T + 3 T^{2} + 2 T^{3} + 4 T^{4}$ $1 + T + 3 T^{2}$ $1 + 5 T + 15 T^{2} + 25 T^{3} + 25 T^{4}$ $1 - T^{2} + 49 T^{4}$
4-25100-1.1-c1e2-0-2 Elliptic curve 2.2.5.1-1004.1-c Elliptic curve 2.2.5.1-1004.2-c Hilbert modular form 2.2.5.1-1004.1-c Hilbert modular form 2.2.5.1-1004.2-c $1 + T^{2}$ $1 + T^{2} + 9 T^{4}$ $1 + T + 5 T^{2}$ $1 - T^{2} + 49 T^{4}$
4-25625-1.1-c1e2-0-5 Elliptic curve 2.2.5.1-1025.1-f Elliptic curve 2.2.5.1-1025.2-f Hilbert modular form 2.2.5.1-1025.1-f Hilbert modular form 2.2.5.1-1025.2-f $( 1 - T + 2 T^{2} )( 1 + T + 2 T^{2} )$ $1 + T^{2} + 9 T^{4}$ $1$ $1 - T^{2} + 49 T^{4}$
4-25721-1.1-c1e2-0-1 Elliptic curve 2.2.17.1-89.1-a Elliptic curve 2.2.17.1-89.2-a Hilbert modular form 2.2.17.1-89.1-a Hilbert modular form 2.2.17.1-89.2-a $( 1 + 2 T^{2} )( 1 + T + 2 T^{2} )$ $1 + T^{2} + 9 T^{4}$ $( 1 - 3 T + 5 T^{2} )( 1 + 3 T + 5 T^{2} )$ $1 - T^{2} + 49 T^{4}$
4-25846-1.1-c1e2-0-0 Genus 2 curve 25846.b $( 1 + T )( 1 + T + 2 T^{2} )$ $1 + 2 T + 5 T^{2} + 6 T^{3} + 9 T^{4}$ $1 - 2 T + 8 T^{2} - 10 T^{3} + 25 T^{4}$ $1 - T^{2} + 49 T^{4}$
4-26100-1.1-c1e2-0-1 Elliptic curve 2.2.5.1-1044.1-b Elliptic curve 2.2.5.1-1044.2-b Hilbert modular form 2.2.5.1-1044.1-b Hilbert modular form 2.2.5.1-1044.2-b $1 + T^{2}$ $1 + T^{2}$ $1 - 2 T + 5 T^{2}$ $1 - T^{2} + 49 T^{4}$
4-26100-1.1-c1e2-0-2 Elliptic curve 2.2.5.1-1044.1-a Elliptic curve 2.2.5.1-1044.2-a Hilbert modular form 2.2.5.1-1044.1-a Hilbert modular form 2.2.5.1-1044.2-a $( 1 - T )( 1 + T )$ $1 + T^{2}$ $1 - 2 T + 5 T^{2}$ $1 - T^{2} + 49 T^{4}$
4-26587-1.1-c1e2-0-0 Genus 2 curve 26587.b $1 + T + 3 T^{2} + 2 T^{3} + 4 T^{4}$ $( 1 + T + 3 T^{2} )( 1 + 3 T + 3 T^{2} )$ $1 + T + 6 T^{2} + 5 T^{3} + 25 T^{4}$ $1 - T^{2} + 49 T^{4}$
4-26900-1.1-c1e2-0-1 Elliptic curve 2.2.5.1-1076.1-b Elliptic curve 2.2.5.1-1076.2-b Hilbert modular form 2.2.5.1-1076.1-b Hilbert modular form 2.2.5.1-1076.2-b $( 1 - T )( 1 + T )$ $1 + 3 T^{2} + 9 T^{4}$ $1 + 3 T + 5 T^{2}$ $1 - T^{2} + 49 T^{4}$
4-27500-1.1-c1e2-0-0 Elliptic curve 2.2.5.1-1100.1-f Elliptic curve 2.2.5.1-1100.2-f Hilbert modular form 2.2.5.1-1100.1-f Hilbert modular form 2.2.5.1-1100.2-f $1 + T^{2}$ $1 + 4 T^{2} + 9 T^{4}$ $1$ $1 - T^{2} + 49 T^{4}$
4-28100-1.1-c1e2-0-0 Elliptic curve 2.2.5.1-1124.1-a Elliptic curve 2.2.5.1-1124.2-a Hilbert modular form 2.2.5.1-1124.1-a Hilbert modular form 2.2.5.1-1124.2-a $( 1 - T )( 1 + T )$ $1 - 2 T^{2} + 9 T^{4}$ $1 + 5 T^{2}$ $1 - T^{2} + 49 T^{4}$
4-29500-1.1-c1e2-0-2 Elliptic curve 2.2.5.1-1180.1-c Elliptic curve 2.2.5.1-1180.2-c Hilbert modular form 2.2.5.1-1180.1-c Hilbert modular form 2.2.5.1-1180.2-c $1 + T^{2}$ $1 - T^{2} + 9 T^{4}$ $1 + T$ $1 - T^{2} + 49 T^{4}$
4-29508-1.1-c1e2-0-0 Genus 2 curve 29508.a $1 + T^{2}$ $( 1 - T )( 1 + 2 T + 3 T^{2} )$ $1 - T + 6 T^{2} - 5 T^{3} + 25 T^{4}$ $1 - T^{2} + 49 T^{4}$
4-29520-1.1-c1e2-0-0 Bianchi modular form 2.0.4.1-1845.2-a Bianchi modular form 2.0.4.1-1845.3-a Elliptic curve 2.0.4.1-1845.2-a Elliptic curve 2.0.4.1-1845.3-a $1 - T + 2 T^{2}$ $1 + T^{2}$ $( 1 + T )( 1 + T + 5 T^{2} )$ $1 - T^{2} + 49 T^{4}$
4-29824-1.1-c1e2-0-1 Bianchi modular form 2.0.4.1-1864.1-a Bianchi modular form 2.0.4.1-1864.2-b Elliptic curve 2.0.4.1-1864.1-a Elliptic curve 2.0.4.1-1864.2-b $1$ $1 + 4 T^{2} + 9 T^{4}$ $( 1 - T + 5 T^{2} )( 1 + 2 T + 5 T^{2} )$ $1 - T^{2} + 49 T^{4}$
4-32575-1.1-c1e2-0-1 Genus 2 curve 32575.b $1 + 3 T + 5 T^{2} + 6 T^{3} + 4 T^{4}$ $( 1 + 3 T^{2} )( 1 + 3 T + 3 T^{2} )$ $( 1 - T )( 1 + T )$ $1 - T^{2} + 49 T^{4}$
4-33275-1.1-c1e2-0-3 Elliptic curve 2.2.5.1-1331.1-d Elliptic curve 2.2.5.1-1331.2-d Hilbert modular form 2.2.5.1-1331.1-d Hilbert modular form 2.2.5.1-1331.2-d $( 1 - T + 2 T^{2} )( 1 + T + 2 T^{2} )$ $1 - T^{2} + 9 T^{4}$ $1 + T + 5 T^{2}$ $1 - T^{2} + 49 T^{4}$
4-34175-1.1-c1e2-0-0 Genus 2 curve 34175.a $1 + 3 T + 5 T^{2} + 6 T^{3} + 4 T^{4}$ $1 + 3 T + 7 T^{2} + 9 T^{3} + 9 T^{4}$ $1 + T + 5 T^{2}$ $1 - T^{2} + 49 T^{4}$
4-34799-1.1-c1e2-0-0 Genus 2 curve 34799.a $1 + T + T^{2} + 2 T^{3} + 4 T^{4}$ $1 + 2 T + 5 T^{2} + 6 T^{3} + 9 T^{4}$ $1 - T - 5 T^{3} + 25 T^{4}$ $1 - T^{2} + 49 T^{4}$
4-34848-1.1-c1e2-0-17 Elliptic curve 2.2.12.1-242.2-b Elliptic curve 2.2.12.1-242.3-b Hilbert modular form 2.2.12.1-242.2-b Hilbert modular form 2.2.12.1-242.3-b $1 - T$ $1 - 3 T + 3 T^{2}$ $1 + 8 T^{2} + 25 T^{4}$ $1 - T^{2} + 49 T^{4}$
4-34848-1.1-c1e2-0-7 Elliptic curve 2.2.12.1-242.2-c Elliptic curve 2.2.12.1-242.3-c Hilbert modular form 2.2.12.1-242.2-c Hilbert modular form 2.2.12.1-242.3-c $1 + T$ $1 + 3 T + 3 T^{2}$ $1 + 8 T^{2} + 25 T^{4}$ $1 - T^{2} + 49 T^{4}$
4-37349-1.1-c1e2-0-0 Elliptic curve 2.2.13.1-221.1-c Elliptic curve 2.2.13.1-221.2-c Hilbert modular form 2.2.13.1-221.1-c Hilbert modular form 2.2.13.1-221.2-c $1 - T^{2} + 4 T^{4}$ $( 1 + 2 T + 3 T^{2} )( 1 + 3 T + 3 T^{2} )$ $1 - 7 T^{2} + 25 T^{4}$ $1 - T^{2} + 49 T^{4}$
4-38875-1.1-c1e2-0-2 Elliptic curve 2.2.5.1-1555.1-b Elliptic curve 2.2.5.1-1555.2-b Hilbert modular form 2.2.5.1-1555.1-b Hilbert modular form 2.2.5.1-1555.2-b $1 + 2 T^{2} + 4 T^{4}$ $1 - T^{2} + 9 T^{4}$ $1 - T$ $1 - T^{2} + 49 T^{4}$
4-41375-1.1-c1e2-0-0 Elliptic curve 2.2.5.1-1655.1-c Elliptic curve 2.2.5.1-1655.2-c Hilbert modular form 2.2.5.1-1655.1-c Hilbert modular form 2.2.5.1-1655.2-c $1 + T^{2} + 4 T^{4}$ $1 + 3 T^{2} + 9 T^{4}$ $1 + T$ $1 - T^{2} + 49 T^{4}$
4-42625-1.1-c1e2-0-3 Elliptic curve 2.2.5.1-1705.1-d Elliptic curve 2.2.5.1-1705.4-d Hilbert modular form 2.2.5.1-1705.1-d Hilbert modular form 2.2.5.1-1705.4-d $( 1 - T + 2 T^{2} )( 1 + T + 2 T^{2} )$ $1 + 3 T^{2} + 9 T^{4}$ $1 - T$ $1 - T^{2} + 49 T^{4}$
4-42724-1.1-c1e2-0-0 Genus 2 curve 42724.a $1 + T + T^{2}$ $( 1 + 3 T^{2} )( 1 + 2 T + 3 T^{2} )$ $1 + 4 T + 9 T^{2} + 20 T^{3} + 25 T^{4}$ $1 - T^{2} + 49 T^{4}$
4-43056-1.1-c1e2-0-1 Elliptic curve 2.2.12.1-299.2-d Elliptic curve 2.2.12.1-299.3-d Hilbert modular form 2.2.12.1-299.2-d Hilbert modular form 2.2.12.1-299.3-d $1 + T + 2 T^{2}$ $1 + 3 T + 3 T^{2}$ $1 - 3 T^{2} + 25 T^{4}$ $1 - T^{2} + 49 T^{4}$
4-43056-1.1-c1e2-0-2 Elliptic curve 2.2.12.1-299.2-a Elliptic curve 2.2.12.1-299.3-a Hilbert modular form 2.2.12.1-299.2-a Hilbert modular form 2.2.12.1-299.3-a $1 - T + 2 T^{2}$ $1 - 3 T + 3 T^{2}$ $1 - 3 T^{2} + 25 T^{4}$ $1 - T^{2} + 49 T^{4}$
4-208e2-1.1-c1e2-0-11 Elliptic curve 2.2.13.1-256.1-e Elliptic curve 2.2.13.1-256.1-h Hilbert modular form 2.2.13.1-256.1-e Hilbert modular form 2.2.13.1-256.1-h $1$ $( 1 - T + 3 T^{2} )( 1 + 3 T + 3 T^{2} )$ $1 + 7 T^{2} + 25 T^{4}$ $1 - T^{2} + 49 T^{4}$
4-44528-1.1-c1e2-0-0 Bianchi modular form 2.0.11.1-368.1-a Bianchi modular form 2.0.11.1-368.2-a Elliptic curve 2.0.11.1-368.1-a Elliptic curve 2.0.11.1-368.2-a $1$ $( 1 - 3 T + 3 T^{2} )( 1 + 3 T^{2} )$ $( 1 - T + 5 T^{2} )( 1 + 5 T^{2} )$ $1 - T^{2} + 49 T^{4}$
4-46240-1.1-c1e2-0-5 Bianchi modular form 2.0.4.1-2890.3-b Bianchi modular form 2.0.4.1-2890.4-b Elliptic curve 2.0.4.1-2890.3-b Elliptic curve 2.0.4.1-2890.4-b $1 - T$ $1 + 4 T^{2} + 9 T^{4}$ $( 1 - T )( 1 + 3 T + 5 T^{2} )$ $1 - T^{2} + 49 T^{4}$
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