Properties

Label 19.2.e.a
Level 19
Weight 2
Character orbit 19.e
Analytic conductor 0.152
Analytic rank 0
Dimension 6
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 19 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 19.e (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.151715763840\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{18}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \zeta_{18} - \zeta_{18}^{2} ) q^{2} + ( -1 + \zeta_{18}^{2} + \zeta_{18}^{3} - \zeta_{18}^{4} ) q^{3} + ( 1 - 2 \zeta_{18} + \zeta_{18}^{2} - 2 \zeta_{18}^{3} + \zeta_{18}^{4} ) q^{4} + ( -1 - \zeta_{18} + \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{5} + ( -\zeta_{18} + \zeta_{18}^{3} + \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{6} + ( \zeta_{18} - \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{7} + ( 3 \zeta_{18} - 2 \zeta_{18}^{2} + 2 \zeta_{18}^{3} - 2 \zeta_{18}^{4} + 3 \zeta_{18}^{5} ) q^{8} + ( 2 + 2 \zeta_{18} - 3 \zeta_{18}^{3} + \zeta_{18}^{4} ) q^{9} +O(q^{10})\) \( q + ( -1 + \zeta_{18} - \zeta_{18}^{2} ) q^{2} + ( -1 + \zeta_{18}^{2} + \zeta_{18}^{3} - \zeta_{18}^{4} ) q^{3} + ( 1 - 2 \zeta_{18} + \zeta_{18}^{2} - 2 \zeta_{18}^{3} + \zeta_{18}^{4} ) q^{4} + ( -1 - \zeta_{18} + \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{5} + ( -\zeta_{18} + \zeta_{18}^{3} + \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{6} + ( \zeta_{18} - \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{7} + ( 3 \zeta_{18} - 2 \zeta_{18}^{2} + 2 \zeta_{18}^{3} - 2 \zeta_{18}^{4} + 3 \zeta_{18}^{5} ) q^{8} + ( 2 + 2 \zeta_{18} - 3 \zeta_{18}^{3} + \zeta_{18}^{4} ) q^{9} + ( 1 + \zeta_{18} + \zeta_{18}^{3} - 2 \zeta_{18}^{4} ) q^{10} + ( -\zeta_{18} - \zeta_{18}^{2} - \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{11} + ( 1 - \zeta_{18} + \zeta_{18}^{2} - \zeta_{18}^{3} ) q^{12} + ( -2 - \zeta_{18} + 3 \zeta_{18}^{3} + 3 \zeta_{18}^{4} ) q^{13} + ( -2 \zeta_{18} + \zeta_{18}^{2} - \zeta_{18}^{3} + 2 \zeta_{18}^{4} ) q^{14} + ( 1 - \zeta_{18} - \zeta_{18}^{2} - \zeta_{18}^{3} + \zeta_{18}^{4} ) q^{15} + ( -3 + 3 \zeta_{18}^{2} + \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{16} + ( 1 - 2 \zeta_{18} - \zeta_{18}^{3} + \zeta_{18}^{5} ) q^{17} + ( -1 - 4 \zeta_{18}^{4} + 4 \zeta_{18}^{5} ) q^{18} + ( -2 + 2 \zeta_{18} - 2 \zeta_{18}^{2} - 4 \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{19} + ( -1 + 2 \zeta_{18} + 2 \zeta_{18}^{2} + \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{20} + \zeta_{18} q^{21} + 3 \zeta_{18}^{4} q^{22} + ( 2 \zeta_{18} - 2 \zeta_{18}^{2} + 2 \zeta_{18}^{3} ) q^{23} + ( 1 - 2 \zeta_{18} - 3 \zeta_{18}^{2} + 3 \zeta_{18}^{3} + 2 \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{24} + ( 1 + \zeta_{18} - 2 \zeta_{18}^{3} - 2 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{25} + ( 5 - \zeta_{18} + \zeta_{18}^{2} - 5 \zeta_{18}^{3} ) q^{26} + ( -3 \zeta_{18} + 3 \zeta_{18}^{2} + 2 \zeta_{18}^{3} + 3 \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{27} + ( -3 \zeta_{18}^{2} + 2 \zeta_{18}^{3} - 2 \zeta_{18}^{4} + 3 \zeta_{18}^{5} ) q^{28} + ( -1 - \zeta_{18} + 5 \zeta_{18}^{2} + \zeta_{18}^{3} - 5 \zeta_{18}^{5} ) q^{29} + ( 2 \zeta_{18} - \zeta_{18}^{2} - \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{30} + ( 3 - \zeta_{18} - 2 \zeta_{18}^{2} - 3 \zeta_{18}^{3} + 3 \zeta_{18}^{4} + 3 \zeta_{18}^{5} ) q^{31} + ( 3 - 3 \zeta_{18}^{3} - 3 \zeta_{18}^{4} ) q^{32} + ( -1 + 3 \zeta_{18} + \zeta_{18}^{2} - \zeta_{18}^{3} - 3 \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{33} + ( -2 + 4 \zeta_{18} - 3 \zeta_{18}^{2} + 4 \zeta_{18}^{3} - 2 \zeta_{18}^{4} ) q^{34} + ( 2 \zeta_{18}^{3} + \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{35} + ( -5 + 3 \zeta_{18} - 3 \zeta_{18}^{2} + 2 \zeta_{18}^{3} - 2 \zeta_{18}^{5} ) q^{36} + ( -\zeta_{18} - \zeta_{18}^{2} + 3 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{37} + ( -3 - 3 \zeta_{18} + 6 \zeta_{18}^{2} + \zeta_{18}^{3} + 5 \zeta_{18}^{4} - 5 \zeta_{18}^{5} ) q^{38} + ( -4 + \zeta_{18} + \zeta_{18}^{2} - 2 \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{39} + ( 1 - 6 \zeta_{18} - \zeta_{18}^{2} - 2 \zeta_{18}^{3} + 2 \zeta_{18}^{5} ) q^{40} + ( 4 - 4 \zeta_{18}^{2} - \zeta_{18}^{3} - 4 \zeta_{18}^{4} + 3 \zeta_{18}^{5} ) q^{41} + ( -\zeta_{18} + \zeta_{18}^{2} - \zeta_{18}^{3} ) q^{42} + ( 2 + 5 \zeta_{18}^{2} - 5 \zeta_{18}^{3} - 2 \zeta_{18}^{5} ) q^{43} + ( 1 - 2 \zeta_{18} + \zeta_{18}^{3} + \zeta_{18}^{4} + 3 \zeta_{18}^{5} ) q^{44} + ( -5 - \zeta_{18} + 5 \zeta_{18}^{3} + \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{45} + ( -2 \zeta_{18} + 4 \zeta_{18}^{2} - 6 \zeta_{18}^{3} + 4 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{46} + ( -2 - 2 \zeta_{18} + 2 \zeta_{18}^{2} + 3 \zeta_{18}^{3} - \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{47} + ( 2 + 2 \zeta_{18} - 2 \zeta_{18}^{2} - 5 \zeta_{18}^{3} + 3 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{48} + ( -\zeta_{18} + 5 \zeta_{18}^{3} - \zeta_{18}^{5} ) q^{49} + ( -5 + 2 \zeta_{18} + 5 \zeta_{18}^{3} - 2 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{50} + ( 1 - \zeta_{18}^{3} - \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{51} + ( 1 + 6 \zeta_{18} - 3 \zeta_{18}^{2} + 3 \zeta_{18}^{3} - 6 \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{52} + ( 1 - 3 \zeta_{18} - 2 \zeta_{18}^{2} - 3 \zeta_{18}^{3} + \zeta_{18}^{4} ) q^{53} + ( 6 - 6 \zeta_{18}^{2} - 2 \zeta_{18}^{3} - \zeta_{18}^{4} + 4 \zeta_{18}^{5} ) q^{54} + ( 3 + 3 \zeta_{18} + 3 \zeta_{18}^{2} ) q^{55} + ( -1 + 3 \zeta_{18} + 3 \zeta_{18}^{2} + 2 \zeta_{18}^{4} - 5 \zeta_{18}^{5} ) q^{56} + ( 5 + \zeta_{18} - 5 \zeta_{18}^{2} - 2 \zeta_{18}^{3} + 3 \zeta_{18}^{4} ) q^{57} + ( 6 - 5 \zeta_{18} - 5 \zeta_{18}^{2} + \zeta_{18}^{4} + 4 \zeta_{18}^{5} ) q^{58} + ( 2 + 7 \zeta_{18} + 2 \zeta_{18}^{2} ) q^{59} + ( -1 + \zeta_{18}^{2} + 2 \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{60} + ( -4 + 4 \zeta_{18} + 3 \zeta_{18}^{2} + 4 \zeta_{18}^{3} - 4 \zeta_{18}^{4} ) q^{61} + ( -3 + 7 \zeta_{18} - 2 \zeta_{18}^{2} + 2 \zeta_{18}^{3} - 7 \zeta_{18}^{4} + 3 \zeta_{18}^{5} ) q^{62} + ( 3 - \zeta_{18} - 2 \zeta_{18}^{3} - 2 \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{63} + ( -4 - 3 \zeta_{18} + 3 \zeta_{18}^{2} + 4 \zeta_{18}^{3} ) q^{64} + ( -5 \zeta_{18}^{2} - 4 \zeta_{18}^{3} - 5 \zeta_{18}^{4} ) q^{65} + ( -3 - 3 \zeta_{18} + 3 \zeta_{18}^{2} + 3 \zeta_{18}^{3} - 3 \zeta_{18}^{5} ) q^{66} + ( -4 - 4 \zeta_{18} - 6 \zeta_{18}^{2} - 2 \zeta_{18}^{3} + 6 \zeta_{18}^{4} + 6 \zeta_{18}^{5} ) q^{67} + ( -4 \zeta_{18} + 7 \zeta_{18}^{2} - 5 \zeta_{18}^{3} + 7 \zeta_{18}^{4} - 4 \zeta_{18}^{5} ) q^{68} + ( -4 + 2 \zeta_{18}^{2} + 4 \zeta_{18}^{3} - 2 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{69} + ( -1 + 2 \zeta_{18} - \zeta_{18}^{3} - \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{70} + ( -2 - 10 \zeta_{18} - 2 \zeta_{18}^{2} + 2 \zeta_{18}^{3} + 10 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{71} + ( -1 - 2 \zeta_{18} + 13 \zeta_{18}^{2} - 2 \zeta_{18}^{3} - \zeta_{18}^{4} ) q^{72} + ( -4 \zeta_{18}^{3} - 4 \zeta_{18}^{5} ) q^{73} + ( 5 - \zeta_{18} - 5 \zeta_{18}^{3} + 5 \zeta_{18}^{5} ) q^{74} + ( 5 - 3 \zeta_{18} - 3 \zeta_{18}^{2} + 4 \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{75} + ( 5 - 3 \zeta_{18} - 2 \zeta_{18}^{2} + 2 \zeta_{18}^{3} - 3 \zeta_{18}^{4} + 9 \zeta_{18}^{5} ) q^{76} + ( -3 - 2 \zeta_{18} - 2 \zeta_{18}^{2} + \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{77} + ( 1 - 4 \zeta_{18} + 4 \zeta_{18}^{2} + 3 \zeta_{18}^{3} - 3 \zeta_{18}^{5} ) q^{78} + ( -6 + 6 \zeta_{18}^{2} - \zeta_{18}^{3} - 3 \zeta_{18}^{4} - 7 \zeta_{18}^{5} ) q^{79} + ( -1 + 3 \zeta_{18} - 4 \zeta_{18}^{2} + 3 \zeta_{18}^{3} - \zeta_{18}^{4} ) q^{80} + ( 1 + 5 \zeta_{18} - 5 \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{81} + ( -11 + 7 \zeta_{18} + 4 \zeta_{18}^{3} + 4 \zeta_{18}^{4} - 6 \zeta_{18}^{5} ) q^{82} + ( 6 \zeta_{18} - 9 \zeta_{18}^{2} + 3 \zeta_{18}^{4} + 3 \zeta_{18}^{5} ) q^{83} + ( \zeta_{18} - 2 \zeta_{18}^{2} + \zeta_{18}^{3} - 2 \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{84} + ( 1 + \zeta_{18} + 3 \zeta_{18}^{2} - 2 \zeta_{18}^{3} + \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{85} + ( -7 \zeta_{18}^{2} + 8 \zeta_{18}^{3} - 8 \zeta_{18}^{4} + 7 \zeta_{18}^{5} ) q^{86} + ( 7 \zeta_{18} - \zeta_{18}^{2} - 7 \zeta_{18}^{3} - \zeta_{18}^{4} + 7 \zeta_{18}^{5} ) q^{87} + ( 3 + 6 \zeta_{18} - 3 \zeta_{18}^{2} - 3 \zeta_{18}^{3} - 3 \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{88} + ( -1 + 3 \zeta_{18} - 2 \zeta_{18}^{3} - 2 \zeta_{18}^{4} + 5 \zeta_{18}^{5} ) q^{89} + ( 5 - 3 \zeta_{18} + 4 \zeta_{18}^{2} - 4 \zeta_{18}^{3} + 3 \zeta_{18}^{4} - 5 \zeta_{18}^{5} ) q^{90} + ( 2 + \zeta_{18} + 5 \zeta_{18}^{2} + \zeta_{18}^{3} + 2 \zeta_{18}^{4} ) q^{91} + ( 6 - 6 \zeta_{18}^{2} + 2 \zeta_{18}^{3} - 8 \zeta_{18}^{4} + 8 \zeta_{18}^{5} ) q^{92} + ( -2 - 8 \zeta_{18} + 5 \zeta_{18}^{2} + 7 \zeta_{18}^{3} - 7 \zeta_{18}^{5} ) q^{93} + ( 3 - 2 \zeta_{18} - 2 \zeta_{18}^{2} + 4 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{94} + ( 6 + \zeta_{18} + 4 \zeta_{18}^{2} + \zeta_{18}^{3} - \zeta_{18}^{5} ) q^{95} + 3 q^{96} + ( 2 - 5 \zeta_{18} + 4 \zeta_{18}^{2} + 2 \zeta_{18}^{3} - 2 \zeta_{18}^{5} ) q^{97} + ( 1 - \zeta_{18}^{2} - 5 \zeta_{18}^{3} + 6 \zeta_{18}^{4} - 4 \zeta_{18}^{5} ) q^{98} + ( 4 - 5 \zeta_{18} - 6 \zeta_{18}^{2} - 5 \zeta_{18}^{3} + 4 \zeta_{18}^{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 6q^{2} - 3q^{3} - 6q^{5} + 3q^{6} + 6q^{8} + 3q^{9} + O(q^{10}) \) \( 6q - 6q^{2} - 3q^{3} - 6q^{5} + 3q^{6} + 6q^{8} + 3q^{9} + 9q^{10} + 3q^{12} - 3q^{13} - 3q^{14} + 3q^{15} - 18q^{16} + 3q^{17} - 6q^{18} - 12q^{19} - 6q^{20} + 6q^{23} + 15q^{24} + 15q^{26} + 6q^{27} + 6q^{28} - 3q^{29} + 9q^{31} + 9q^{32} - 9q^{33} + 6q^{35} - 24q^{36} - 15q^{38} - 24q^{39} + 21q^{41} - 3q^{42} - 3q^{43} + 9q^{44} - 15q^{45} - 18q^{46} - 3q^{47} - 3q^{48} + 15q^{49} - 15q^{50} + 3q^{51} + 15q^{52} - 3q^{53} + 30q^{54} + 18q^{55} - 6q^{56} + 24q^{57} + 36q^{58} + 12q^{59} - 6q^{60} - 12q^{61} - 12q^{62} + 12q^{63} - 12q^{64} - 12q^{65} - 9q^{66} - 30q^{67} - 15q^{68} - 12q^{69} - 9q^{70} - 6q^{71} - 12q^{72} - 12q^{73} + 15q^{74} + 30q^{75} + 36q^{76} - 18q^{77} + 15q^{78} - 39q^{79} + 3q^{80} + 6q^{81} - 54q^{82} + 3q^{84} + 24q^{86} - 21q^{87} + 9q^{88} - 12q^{89} + 18q^{90} + 15q^{91} + 42q^{92} + 9q^{93} + 18q^{94} + 39q^{95} + 18q^{96} + 18q^{97} - 9q^{98} + 9q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/19\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(\zeta_{18}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
0.939693 + 0.342020i
0.939693 0.342020i
−0.766044 + 0.642788i
−0.173648 0.984808i
−0.766044 0.642788i
−0.173648 + 0.984808i
−0.826352 0.300767i 0.0923963 + 0.524005i −0.939693 0.788496i −1.93969 + 1.62760i 0.0812519 0.460802i 0.939693 1.62760i 1.41875 + 2.45734i 2.55303 0.929228i 2.09240 0.761570i
5.1 −0.826352 + 0.300767i 0.0923963 0.524005i −0.939693 + 0.788496i −1.93969 1.62760i 0.0812519 + 0.460802i 0.939693 + 1.62760i 1.41875 2.45734i 2.55303 + 0.929228i 2.09240 + 0.761570i
6.1 −1.93969 + 1.62760i 0.613341 + 0.223238i 0.766044 4.34445i −0.233956 1.32683i −1.55303 + 0.565258i −0.766044 + 1.32683i 3.05303 + 5.28801i −1.97178 1.65452i 2.61334 + 2.19285i
9.1 −0.233956 1.32683i −2.20574 + 1.85083i 0.173648 0.0632028i −0.826352 0.300767i 2.97178 + 2.49362i −0.173648 + 0.300767i −1.47178 2.54920i 0.918748 5.21048i −0.205737 + 1.16679i
16.1 −1.93969 1.62760i 0.613341 0.223238i 0.766044 + 4.34445i −0.233956 + 1.32683i −1.55303 0.565258i −0.766044 1.32683i 3.05303 5.28801i −1.97178 + 1.65452i 2.61334 2.19285i
17.1 −0.233956 + 1.32683i −2.20574 1.85083i 0.173648 + 0.0632028i −0.826352 + 0.300767i 2.97178 2.49362i −0.173648 0.300767i −1.47178 + 2.54920i 0.918748 + 5.21048i −0.205737 1.16679i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 19.2.e.a 6
3.b odd 2 1 171.2.u.c 6
4.b odd 2 1 304.2.u.b 6
5.b even 2 1 475.2.l.a 6
5.c odd 4 2 475.2.u.a 12
7.b odd 2 1 931.2.w.a 6
7.c even 3 1 931.2.v.b 6
7.c even 3 1 931.2.x.a 6
7.d odd 6 1 931.2.v.a 6
7.d odd 6 1 931.2.x.b 6
19.b odd 2 1 361.2.e.h 6
19.c even 3 1 361.2.e.f 6
19.c even 3 1 361.2.e.g 6
19.d odd 6 1 361.2.e.a 6
19.d odd 6 1 361.2.e.b 6
19.e even 9 1 inner 19.2.e.a 6
19.e even 9 1 361.2.a.g 3
19.e even 9 2 361.2.c.i 6
19.e even 9 1 361.2.e.f 6
19.e even 9 1 361.2.e.g 6
19.f odd 18 1 361.2.a.h 3
19.f odd 18 2 361.2.c.h 6
19.f odd 18 1 361.2.e.a 6
19.f odd 18 1 361.2.e.b 6
19.f odd 18 1 361.2.e.h 6
57.j even 18 1 3249.2.a.s 3
57.l odd 18 1 171.2.u.c 6
57.l odd 18 1 3249.2.a.z 3
76.k even 18 1 5776.2.a.bi 3
76.l odd 18 1 304.2.u.b 6
76.l odd 18 1 5776.2.a.br 3
95.o odd 18 1 9025.2.a.x 3
95.p even 18 1 475.2.l.a 6
95.p even 18 1 9025.2.a.bd 3
95.q odd 36 2 475.2.u.a 12
133.u even 9 1 931.2.x.a 6
133.w even 9 1 931.2.v.b 6
133.x odd 18 1 931.2.x.b 6
133.y odd 18 1 931.2.w.a 6
133.z odd 18 1 931.2.v.a 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.2.e.a 6 1.a even 1 1 trivial
19.2.e.a 6 19.e even 9 1 inner
171.2.u.c 6 3.b odd 2 1
171.2.u.c 6 57.l odd 18 1
304.2.u.b 6 4.b odd 2 1
304.2.u.b 6 76.l odd 18 1
361.2.a.g 3 19.e even 9 1
361.2.a.h 3 19.f odd 18 1
361.2.c.h 6 19.f odd 18 2
361.2.c.i 6 19.e even 9 2
361.2.e.a 6 19.d odd 6 1
361.2.e.a 6 19.f odd 18 1
361.2.e.b 6 19.d odd 6 1
361.2.e.b 6 19.f odd 18 1
361.2.e.f 6 19.c even 3 1
361.2.e.f 6 19.e even 9 1
361.2.e.g 6 19.c even 3 1
361.2.e.g 6 19.e even 9 1
361.2.e.h 6 19.b odd 2 1
361.2.e.h 6 19.f odd 18 1
475.2.l.a 6 5.b even 2 1
475.2.l.a 6 95.p even 18 1
475.2.u.a 12 5.c odd 4 2
475.2.u.a 12 95.q odd 36 2
931.2.v.a 6 7.d odd 6 1
931.2.v.a 6 133.z odd 18 1
931.2.v.b 6 7.c even 3 1
931.2.v.b 6 133.w even 9 1
931.2.w.a 6 7.b odd 2 1
931.2.w.a 6 133.y odd 18 1
931.2.x.a 6 7.c even 3 1
931.2.x.a 6 133.u even 9 1
931.2.x.b 6 7.d odd 6 1
931.2.x.b 6 133.x odd 18 1
3249.2.a.s 3 57.j even 18 1
3249.2.a.z 3 57.l odd 18 1
5776.2.a.bi 3 76.k even 18 1
5776.2.a.br 3 76.l odd 18 1
9025.2.a.x 3 95.o odd 18 1
9025.2.a.bd 3 95.p even 18 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(19, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 6 T + 18 T^{2} + 36 T^{3} + 54 T^{4} + 69 T^{5} + 91 T^{6} + 138 T^{7} + 216 T^{8} + 288 T^{9} + 288 T^{10} + 192 T^{11} + 64 T^{12} \)
$3$ \( 1 + 3 T + 3 T^{2} + T^{3} - 12 T^{4} - 30 T^{5} - 35 T^{6} - 90 T^{7} - 108 T^{8} + 27 T^{9} + 243 T^{10} + 729 T^{11} + 729 T^{12} \)
$5$ \( 1 + 6 T + 18 T^{2} + 45 T^{3} + 81 T^{4} + 87 T^{5} + 109 T^{6} + 435 T^{7} + 2025 T^{8} + 5625 T^{9} + 11250 T^{10} + 18750 T^{11} + 15625 T^{12} \)
$7$ \( 1 - 18 T^{2} + 2 T^{3} + 198 T^{4} - 18 T^{5} - 1581 T^{6} - 126 T^{7} + 9702 T^{8} + 686 T^{9} - 43218 T^{10} + 117649 T^{12} \)
$11$ \( 1 - 24 T^{2} - 18 T^{3} + 312 T^{4} + 216 T^{5} - 3593 T^{6} + 2376 T^{7} + 37752 T^{8} - 23958 T^{9} - 351384 T^{10} + 1771561 T^{12} \)
$13$ \( 1 + 3 T + 24 T^{2} + 26 T^{3} + 315 T^{4} + 261 T^{5} + 4905 T^{6} + 3393 T^{7} + 53235 T^{8} + 57122 T^{9} + 685464 T^{10} + 1113879 T^{11} + 4826809 T^{12} \)
$17$ \( 1 - 3 T - 72 T^{3} - 117 T^{4} + 1275 T^{5} + 1369 T^{6} + 21675 T^{7} - 33813 T^{8} - 353736 T^{9} - 4259571 T^{11} + 24137569 T^{12} \)
$19$ \( 1 + 12 T + 78 T^{2} + 385 T^{3} + 1482 T^{4} + 4332 T^{5} + 6859 T^{6} \)
$23$ \( 1 - 6 T + 36 T^{2} - 54 T^{3} + 576 T^{4} + 516 T^{5} + 4969 T^{6} + 11868 T^{7} + 304704 T^{8} - 657018 T^{9} + 10074276 T^{10} - 38618058 T^{11} + 148035889 T^{12} \)
$29$ \( 1 + 3 T + 36 T^{2} + 378 T^{3} + 1872 T^{4} + 9921 T^{5} + 94159 T^{6} + 287709 T^{7} + 1574352 T^{8} + 9219042 T^{9} + 25462116 T^{10} + 61533447 T^{11} + 594823321 T^{12} \)
$31$ \( 1 - 9 T - 18 T^{2} + 119 T^{3} + 2187 T^{4} - 3402 T^{5} - 67065 T^{6} - 105462 T^{7} + 2101707 T^{8} + 3545129 T^{9} - 16623378 T^{10} - 257662359 T^{11} + 887503681 T^{12} \)
$37$ \( ( 1 + 90 T^{2} - 17 T^{3} + 3330 T^{4} + 50653 T^{6} )^{2} \)
$41$ \( 1 - 21 T + 162 T^{2} - 180 T^{3} - 4707 T^{4} + 28401 T^{5} - 103463 T^{6} + 1164441 T^{7} - 7912467 T^{8} - 12405780 T^{9} + 457773282 T^{10} - 2432980221 T^{11} + 4750104241 T^{12} \)
$43$ \( 1 + 3 T + 60 T^{2} + 8 T^{3} + 2691 T^{4} + 2799 T^{5} + 147141 T^{6} + 120357 T^{7} + 4975659 T^{8} + 636056 T^{9} + 205128060 T^{10} + 441025329 T^{11} + 6321363049 T^{12} \)
$47$ \( 1 + 3 T + 54 T^{2} + 306 T^{3} + 4635 T^{4} + 25239 T^{5} + 190171 T^{6} + 1186233 T^{7} + 10238715 T^{8} + 31769838 T^{9} + 263502774 T^{10} + 688035021 T^{11} + 10779215329 T^{12} \)
$53$ \( 1 + 3 T + 234 T^{3} - 1521 T^{4} - 27771 T^{5} - 29411 T^{6} - 1471863 T^{7} - 4272489 T^{8} + 34837218 T^{9} + 1254586479 T^{11} + 22164361129 T^{12} \)
$59$ \( 1 - 12 T + 18 T^{2} + 1080 T^{3} - 7614 T^{4} - 29982 T^{5} + 754273 T^{6} - 1768938 T^{7} - 26504334 T^{8} + 221809320 T^{9} + 218112498 T^{10} - 8579091588 T^{11} + 42180533641 T^{12} \)
$61$ \( 1 + 12 T + 24 T^{2} - 586 T^{3} - 4140 T^{4} + 44676 T^{5} + 736335 T^{6} + 2725236 T^{7} - 15404940 T^{8} - 133010866 T^{9} + 332300184 T^{10} + 10135155612 T^{11} + 51520374361 T^{12} \)
$67$ \( 1 + 30 T + 348 T^{2} + 1322 T^{3} - 6408 T^{4} - 55008 T^{5} - 101691 T^{6} - 3685536 T^{7} - 28765512 T^{8} + 397608686 T^{9} + 7012590108 T^{10} + 40503753210 T^{11} + 90458382169 T^{12} \)
$71$ \( 1 + 6 T - 36 T^{2} + 594 T^{3} + 3240 T^{4} + 14892 T^{5} + 665785 T^{6} + 1057332 T^{7} + 16332840 T^{8} + 212599134 T^{9} - 914820516 T^{10} + 10825376106 T^{11} + 128100283921 T^{12} \)
$73$ \( 1 + 12 T + 96 T^{2} + 512 T^{3} + 7776 T^{4} + 102924 T^{5} + 981639 T^{6} + 7513452 T^{7} + 41438304 T^{8} + 199176704 T^{9} + 2726231136 T^{10} + 24876859116 T^{11} + 151334226289 T^{12} \)
$79$ \( 1 + 39 T + 708 T^{2} + 8198 T^{3} + 67068 T^{4} + 403623 T^{5} + 2596617 T^{6} + 31886217 T^{7} + 418571388 T^{8} + 4041933722 T^{9} + 27576657348 T^{10} + 120005199561 T^{11} + 243087455521 T^{12} \)
$83$ \( 1 - 60 T^{2} + 918 T^{3} - 1380 T^{4} - 27540 T^{5} + 1055455 T^{6} - 2285820 T^{7} - 9506820 T^{8} + 524900466 T^{9} - 2847499260 T^{10} + 326940373369 T^{12} \)
$89$ \( 1 + 12 T + 54 T^{2} + 1035 T^{3} - 279 T^{4} - 69891 T^{5} + 3961 T^{6} - 6220299 T^{7} - 2209959 T^{8} + 729642915 T^{9} + 3388081014 T^{10} + 67008713388 T^{11} + 496981290961 T^{12} \)
$97$ \( 1 - 18 T + 234 T^{2} - 3310 T^{3} + 39204 T^{4} - 371520 T^{5} + 3748107 T^{6} - 36037440 T^{7} + 368870436 T^{8} - 3020947630 T^{9} + 20715851754 T^{10} - 154572124626 T^{11} + 832972004929 T^{12} \)
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