Properties

Label 19.4.c.a
Level 19
Weight 4
Character orbit 19.c
Analytic conductor 1.121
Analytic rank 0
Dimension 4
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 19 \)
Weight: \( k \) = \( 4 \)
Character orbit: \([\chi]\) = 19.c (of order \(3\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(1.12103629011\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{55})\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + \beta_{2} q^{2} \) \( + ( \beta_{1} + \beta_{3} ) q^{3} \) \( + ( 7 + 7 \beta_{2} ) q^{4} \) \( + ( -\beta_{1} - 7 \beta_{2} - \beta_{3} ) q^{5} \) \( -\beta_{1} q^{6} \) \( + ( -7 - \beta_{3} ) q^{7} \) \( -15 q^{8} \) \( + ( -28 - 28 \beta_{2} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + \beta_{2} q^{2} \) \( + ( \beta_{1} + \beta_{3} ) q^{3} \) \( + ( 7 + 7 \beta_{2} ) q^{4} \) \( + ( -\beta_{1} - 7 \beta_{2} - \beta_{3} ) q^{5} \) \( -\beta_{1} q^{6} \) \( + ( -7 - \beta_{3} ) q^{7} \) \( -15 q^{8} \) \( + ( -28 - 28 \beta_{2} ) q^{9} \) \( + ( 7 + \beta_{1} + 7 \beta_{2} ) q^{10} \) \( + ( 14 - 7 \beta_{3} ) q^{11} \) \( + 7 \beta_{3} q^{12} \) \( + ( 14 - 8 \beta_{1} + 14 \beta_{2} ) q^{13} \) \( + ( \beta_{1} - 7 \beta_{2} + \beta_{3} ) q^{14} \) \( + ( 55 + 7 \beta_{1} + 55 \beta_{2} ) q^{15} \) \( + 41 \beta_{2} q^{16} \) \( + ( 8 \beta_{1} - 56 \beta_{2} + 8 \beta_{3} ) q^{17} \) \( + 28 q^{18} \) \( + ( -70 + 7 \beta_{1} - 42 \beta_{2} + 8 \beta_{3} ) q^{19} \) \( + ( 49 - 7 \beta_{3} ) q^{20} \) \( + ( -7 \beta_{1} + 55 \beta_{2} - 7 \beta_{3} ) q^{21} \) \( + ( 7 \beta_{1} + 14 \beta_{2} + 7 \beta_{3} ) q^{22} \) \( + ( -57 - 7 \beta_{1} - 57 \beta_{2} ) q^{23} \) \( + ( -15 \beta_{1} - 15 \beta_{3} ) q^{24} \) \( + ( 21 - 14 \beta_{1} + 21 \beta_{2} ) q^{25} \) \( + ( -14 - 8 \beta_{3} ) q^{26} \) \( -\beta_{3} q^{27} \) \( + ( -49 + 7 \beta_{1} - 49 \beta_{2} ) q^{28} \) \( + ( -111 + 7 \beta_{1} - 111 \beta_{2} ) q^{29} \) \( + ( -55 + 7 \beta_{3} ) q^{30} \) \( + ( 133 - 7 \beta_{3} ) q^{31} \) \( + ( -161 - 161 \beta_{2} ) q^{32} \) \( + ( 14 \beta_{1} + 385 \beta_{2} + 14 \beta_{3} ) q^{33} \) \( + ( 56 - 8 \beta_{1} + 56 \beta_{2} ) q^{34} \) \( -6 \beta_{2} q^{35} \) \( -196 \beta_{2} q^{36} \) \( + ( 91 - 7 \beta_{3} ) q^{37} \) \( + ( 42 - 8 \beta_{1} - 28 \beta_{2} - \beta_{3} ) q^{38} \) \( + ( 440 + 14 \beta_{3} ) q^{39} \) \( + ( 15 \beta_{1} + 105 \beta_{2} + 15 \beta_{3} ) q^{40} \) \( + ( -34 \beta_{1} + 77 \beta_{2} - 34 \beta_{3} ) q^{41} \) \( + ( -55 + 7 \beta_{1} - 55 \beta_{2} ) q^{42} \) \( + ( 42 \beta_{1} + 134 \beta_{2} + 42 \beta_{3} ) q^{43} \) \( + ( 98 + 49 \beta_{1} + 98 \beta_{2} ) q^{44} \) \( + ( -196 + 28 \beta_{3} ) q^{45} \) \( + ( 57 - 7 \beta_{3} ) q^{46} \) \( + ( -63 - 15 \beta_{1} - 63 \beta_{2} ) q^{47} \) \( -41 \beta_{1} q^{48} \) \( + ( -239 + 14 \beta_{3} ) q^{49} \) \( + ( -21 - 14 \beta_{3} ) q^{50} \) \( + ( -440 + 56 \beta_{1} - 440 \beta_{2} ) q^{51} \) \( + ( -56 \beta_{1} + 98 \beta_{2} - 56 \beta_{3} ) q^{52} \) \( + ( 442 - 28 \beta_{1} + 442 \beta_{2} ) q^{53} \) \( + ( \beta_{1} + \beta_{3} ) q^{54} \) \( + ( -63 \beta_{1} - 483 \beta_{2} - 63 \beta_{3} ) q^{55} \) \( + ( 105 + 15 \beta_{3} ) q^{56} \) \( + ( -385 - 28 \beta_{1} - 440 \beta_{2} - 70 \beta_{3} ) q^{57} \) \( + ( 111 + 7 \beta_{3} ) q^{58} \) \( + ( 13 \beta_{1} + 56 \beta_{2} + 13 \beta_{3} ) q^{59} \) \( + ( 49 \beta_{1} + 385 \beta_{2} + 49 \beta_{3} ) q^{60} \) \( + ( -273 + 29 \beta_{1} - 273 \beta_{2} ) q^{61} \) \( + ( 7 \beta_{1} + 133 \beta_{2} + 7 \beta_{3} ) q^{62} \) \( + ( 196 - 28 \beta_{1} + 196 \beta_{2} ) q^{63} \) \( -167 q^{64} \) \( + ( -342 + 42 \beta_{3} ) q^{65} \) \( + ( -385 - 14 \beta_{1} - 385 \beta_{2} ) q^{66} \) \( + ( 370 + 63 \beta_{1} + 370 \beta_{2} ) q^{67} \) \( + ( 392 + 56 \beta_{3} ) q^{68} \) \( + ( 385 - 57 \beta_{3} ) q^{69} \) \( + ( 6 + 6 \beta_{2} ) q^{70} \) \( + ( 42 \beta_{1} + 216 \beta_{2} + 42 \beta_{3} ) q^{71} \) \( + ( 420 + 420 \beta_{2} ) q^{72} \) \( + ( 6 \beta_{1} + 175 \beta_{2} + 6 \beta_{3} ) q^{73} \) \( + ( 7 \beta_{1} + 91 \beta_{2} + 7 \beta_{3} ) q^{74} \) \( + ( 770 + 21 \beta_{3} ) q^{75} \) \( + ( -196 - 7 \beta_{1} - 490 \beta_{2} + 49 \beta_{3} ) q^{76} \) \( + ( 287 + 35 \beta_{3} ) q^{77} \) \( + ( -14 \beta_{1} + 440 \beta_{2} - 14 \beta_{3} ) q^{78} \) \( + ( -56 \beta_{1} - 76 \beta_{2} - 56 \beta_{3} ) q^{79} \) \( + ( 287 + 41 \beta_{1} + 287 \beta_{2} ) q^{80} \) \( -701 \beta_{2} q^{81} \) \( + ( -77 + 34 \beta_{1} - 77 \beta_{2} ) q^{82} \) \( + ( -952 - 35 \beta_{3} ) q^{83} \) \( + ( -385 - 49 \beta_{3} ) q^{84} \) \( + ( 48 + 48 \beta_{2} ) q^{85} \) \( + ( -134 - 42 \beta_{1} - 134 \beta_{2} ) q^{86} \) \( + ( -385 - 111 \beta_{3} ) q^{87} \) \( + ( -210 + 105 \beta_{3} ) q^{88} \) \( + ( -56 - 104 \beta_{1} - 56 \beta_{2} ) q^{89} \) \( + ( -28 \beta_{1} - 196 \beta_{2} - 28 \beta_{3} ) q^{90} \) \( + ( -538 + 70 \beta_{1} - 538 \beta_{2} ) q^{91} \) \( + ( -49 \beta_{1} - 399 \beta_{2} - 49 \beta_{3} ) q^{92} \) \( + ( 133 \beta_{1} + 385 \beta_{2} + 133 \beta_{3} ) q^{93} \) \( + ( 63 - 15 \beta_{3} ) q^{94} \) \( + ( 91 + 84 \beta_{1} + 636 \beta_{2} + 77 \beta_{3} ) q^{95} \) \( -161 \beta_{3} q^{96} \) \( + ( -62 \beta_{1} - 273 \beta_{2} - 62 \beta_{3} ) q^{97} \) \( + ( -14 \beta_{1} - 239 \beta_{2} - 14 \beta_{3} ) q^{98} \) \( + ( -392 - 196 \beta_{1} - 392 \beta_{2} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut -\mathstrut 2q^{2} \) \(\mathstrut +\mathstrut 14q^{4} \) \(\mathstrut +\mathstrut 14q^{5} \) \(\mathstrut -\mathstrut 28q^{7} \) \(\mathstrut -\mathstrut 60q^{8} \) \(\mathstrut -\mathstrut 56q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut -\mathstrut 2q^{2} \) \(\mathstrut +\mathstrut 14q^{4} \) \(\mathstrut +\mathstrut 14q^{5} \) \(\mathstrut -\mathstrut 28q^{7} \) \(\mathstrut -\mathstrut 60q^{8} \) \(\mathstrut -\mathstrut 56q^{9} \) \(\mathstrut +\mathstrut 14q^{10} \) \(\mathstrut +\mathstrut 56q^{11} \) \(\mathstrut +\mathstrut 28q^{13} \) \(\mathstrut +\mathstrut 14q^{14} \) \(\mathstrut +\mathstrut 110q^{15} \) \(\mathstrut -\mathstrut 82q^{16} \) \(\mathstrut +\mathstrut 112q^{17} \) \(\mathstrut +\mathstrut 112q^{18} \) \(\mathstrut -\mathstrut 196q^{19} \) \(\mathstrut +\mathstrut 196q^{20} \) \(\mathstrut -\mathstrut 110q^{21} \) \(\mathstrut -\mathstrut 28q^{22} \) \(\mathstrut -\mathstrut 114q^{23} \) \(\mathstrut +\mathstrut 42q^{25} \) \(\mathstrut -\mathstrut 56q^{26} \) \(\mathstrut -\mathstrut 98q^{28} \) \(\mathstrut -\mathstrut 222q^{29} \) \(\mathstrut -\mathstrut 220q^{30} \) \(\mathstrut +\mathstrut 532q^{31} \) \(\mathstrut -\mathstrut 322q^{32} \) \(\mathstrut -\mathstrut 770q^{33} \) \(\mathstrut +\mathstrut 112q^{34} \) \(\mathstrut +\mathstrut 12q^{35} \) \(\mathstrut +\mathstrut 392q^{36} \) \(\mathstrut +\mathstrut 364q^{37} \) \(\mathstrut +\mathstrut 224q^{38} \) \(\mathstrut +\mathstrut 1760q^{39} \) \(\mathstrut -\mathstrut 210q^{40} \) \(\mathstrut -\mathstrut 154q^{41} \) \(\mathstrut -\mathstrut 110q^{42} \) \(\mathstrut -\mathstrut 268q^{43} \) \(\mathstrut +\mathstrut 196q^{44} \) \(\mathstrut -\mathstrut 784q^{45} \) \(\mathstrut +\mathstrut 228q^{46} \) \(\mathstrut -\mathstrut 126q^{47} \) \(\mathstrut -\mathstrut 956q^{49} \) \(\mathstrut -\mathstrut 84q^{50} \) \(\mathstrut -\mathstrut 880q^{51} \) \(\mathstrut -\mathstrut 196q^{52} \) \(\mathstrut +\mathstrut 884q^{53} \) \(\mathstrut +\mathstrut 966q^{55} \) \(\mathstrut +\mathstrut 420q^{56} \) \(\mathstrut -\mathstrut 660q^{57} \) \(\mathstrut +\mathstrut 444q^{58} \) \(\mathstrut -\mathstrut 112q^{59} \) \(\mathstrut -\mathstrut 770q^{60} \) \(\mathstrut -\mathstrut 546q^{61} \) \(\mathstrut -\mathstrut 266q^{62} \) \(\mathstrut +\mathstrut 392q^{63} \) \(\mathstrut -\mathstrut 668q^{64} \) \(\mathstrut -\mathstrut 1368q^{65} \) \(\mathstrut -\mathstrut 770q^{66} \) \(\mathstrut +\mathstrut 740q^{67} \) \(\mathstrut +\mathstrut 1568q^{68} \) \(\mathstrut +\mathstrut 1540q^{69} \) \(\mathstrut +\mathstrut 12q^{70} \) \(\mathstrut -\mathstrut 432q^{71} \) \(\mathstrut +\mathstrut 840q^{72} \) \(\mathstrut -\mathstrut 350q^{73} \) \(\mathstrut -\mathstrut 182q^{74} \) \(\mathstrut +\mathstrut 3080q^{75} \) \(\mathstrut +\mathstrut 196q^{76} \) \(\mathstrut +\mathstrut 1148q^{77} \) \(\mathstrut -\mathstrut 880q^{78} \) \(\mathstrut +\mathstrut 152q^{79} \) \(\mathstrut +\mathstrut 574q^{80} \) \(\mathstrut +\mathstrut 1402q^{81} \) \(\mathstrut -\mathstrut 154q^{82} \) \(\mathstrut -\mathstrut 3808q^{83} \) \(\mathstrut -\mathstrut 1540q^{84} \) \(\mathstrut +\mathstrut 96q^{85} \) \(\mathstrut -\mathstrut 268q^{86} \) \(\mathstrut -\mathstrut 1540q^{87} \) \(\mathstrut -\mathstrut 840q^{88} \) \(\mathstrut -\mathstrut 112q^{89} \) \(\mathstrut +\mathstrut 392q^{90} \) \(\mathstrut -\mathstrut 1076q^{91} \) \(\mathstrut +\mathstrut 798q^{92} \) \(\mathstrut -\mathstrut 770q^{93} \) \(\mathstrut +\mathstrut 252q^{94} \) \(\mathstrut -\mathstrut 908q^{95} \) \(\mathstrut +\mathstrut 546q^{97} \) \(\mathstrut +\mathstrut 478q^{98} \) \(\mathstrut -\mathstrut 784q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4}\mathstrut +\mathstrut \) \(55\) \(x^{2}\mathstrut +\mathstrut \) \(3025\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/55\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/55\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(55\) \(\beta_{2}\)
\(\nu^{3}\)\(=\)\(55\) \(\beta_{3}\)

Character Values

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

\(n\) \(2\)
\(\chi(n)\) \(-1 - \beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
3.70810 6.42262i
−3.70810 + 6.42262i
3.70810 + 6.42262i
−3.70810 6.42262i
−0.500000 0.866025i −3.70810 6.42262i 3.50000 6.06218i 7.20810 + 12.4848i −3.70810 + 6.42262i 0.416198 −15.0000 −14.0000 + 24.2487i 7.20810 12.4848i
7.2 −0.500000 0.866025i 3.70810 + 6.42262i 3.50000 6.06218i −0.208099 0.360438i 3.70810 6.42262i −14.4162 −15.0000 −14.0000 + 24.2487i −0.208099 + 0.360438i
11.1 −0.500000 + 0.866025i −3.70810 + 6.42262i 3.50000 + 6.06218i 7.20810 12.4848i −3.70810 6.42262i 0.416198 −15.0000 −14.0000 24.2487i 7.20810 + 12.4848i
11.2 −0.500000 + 0.866025i 3.70810 6.42262i 3.50000 + 6.06218i −0.208099 + 0.360438i 3.70810 + 6.42262i −14.4162 −15.0000 −14.0000 24.2487i −0.208099 0.360438i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
19.c Even 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{2} \) \(\mathstrut +\mathstrut T_{2} \) \(\mathstrut +\mathstrut 1 \) acting on \(S_{4}^{\mathrm{new}}(19, [\chi])\).