Properties

Label 19.4.c.b
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: \(\Q(\sqrt{-3}, \sqrt{73})\)
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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} -\beta_{2} q^{3} + ( -11 + \beta_{1} + 10 \beta_{2} + \beta_{3} ) q^{4} + ( -\beta_{1} - 9 \beta_{2} ) q^{5} + ( -1 + \beta_{1} + \beta_{3} ) q^{6} + ( 12 - 4 \beta_{3} ) q^{7} + ( 21 - 3 \beta_{3} ) q^{8} + ( 26 - 26 \beta_{2} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{2} -\beta_{2} q^{3} + ( -11 + \beta_{1} + 10 \beta_{2} + \beta_{3} ) q^{4} + ( -\beta_{1} - 9 \beta_{2} ) q^{5} + ( -1 + \beta_{1} + \beta_{3} ) q^{6} + ( 12 - 4 \beta_{3} ) q^{7} + ( 21 - 3 \beta_{3} ) q^{8} + ( 26 - 26 \beta_{2} ) q^{9} + ( -28 + 10 \beta_{1} + 18 \beta_{2} + 10 \beta_{3} ) q^{10} + ( -15 - 3 \beta_{3} ) q^{11} + ( 11 - \beta_{3} ) q^{12} + ( -48 - 5 \beta_{1} + 53 \beta_{2} - 5 \beta_{3} ) q^{13} + ( -12 \beta_{1} - 72 \beta_{2} ) q^{14} + ( -10 + \beta_{1} + 9 \beta_{2} + \beta_{3} ) q^{15} + ( -13 \beta_{1} + 26 \beta_{2} ) q^{16} + ( 21 \beta_{1} + 27 \beta_{2} ) q^{17} + ( -26 + 26 \beta_{3} ) q^{18} + ( 19 - 19 \beta_{1} ) q^{19} + ( 128 - 20 \beta_{3} ) q^{20} + ( -4 \beta_{1} - 8 \beta_{2} ) q^{21} + ( 15 \beta_{1} - 54 \beta_{2} ) q^{22} + ( 8 - 17 \beta_{1} + 9 \beta_{2} - 17 \beta_{3} ) q^{23} + ( -3 \beta_{1} - 18 \beta_{2} ) q^{24} + ( 7 + 19 \beta_{1} - 26 \beta_{2} + 19 \beta_{3} ) q^{25} + ( -42 - 48 \beta_{3} ) q^{26} -53 q^{27} + ( -204 + 52 \beta_{1} + 152 \beta_{2} + 52 \beta_{3} ) q^{28} + ( 22 + 41 \beta_{1} - 63 \beta_{2} + 41 \beta_{3} ) q^{29} + ( 28 - 10 \beta_{3} ) q^{30} + ( 14 - 6 \beta_{3} ) q^{31} + ( -53 - 37 \beta_{1} + 90 \beta_{2} - 37 \beta_{3} ) q^{32} + ( -3 \beta_{1} + 18 \beta_{2} ) q^{33} + ( 426 - 48 \beta_{1} - 378 \beta_{2} - 48 \beta_{3} ) q^{34} + ( -48 \beta_{1} - 144 \beta_{2} ) q^{35} + ( 26 \beta_{1} + 260 \beta_{2} ) q^{36} + ( 206 + 36 \beta_{3} ) q^{37} + ( -361 + 342 \beta_{2} + 19 \beta_{3} ) q^{38} + ( 48 + 5 \beta_{3} ) q^{39} + ( -48 \beta_{1} - 216 \beta_{2} ) q^{40} + ( 2 \beta_{1} - 63 \beta_{2} ) q^{41} + ( -84 + 12 \beta_{1} + 72 \beta_{2} + 12 \beta_{3} ) q^{42} + ( 21 \beta_{1} + 145 \beta_{2} ) q^{43} + ( 111 + 15 \beta_{1} - 126 \beta_{2} + 15 \beta_{3} ) q^{44} + ( -260 + 26 \beta_{3} ) q^{45} + ( -314 + 8 \beta_{3} ) q^{46} + ( -186 - 39 \beta_{1} + 225 \beta_{2} - 39 \beta_{3} ) q^{47} + ( 13 + 13 \beta_{1} - 26 \beta_{2} + 13 \beta_{3} ) q^{48} + ( 89 - 80 \beta_{3} ) q^{49} + ( 335 + 7 \beta_{3} ) q^{50} + ( 48 - 21 \beta_{1} - 27 \beta_{2} - 21 \beta_{3} ) q^{51} + ( 2 \beta_{1} - 440 \beta_{2} ) q^{52} + ( -180 + 99 \beta_{1} + 81 \beta_{2} + 99 \beta_{3} ) q^{53} + 53 \beta_{1} q^{54} + ( -12 \beta_{1} + 108 \beta_{2} ) q^{55} + ( 468 - 108 \beta_{3} ) q^{56} + ( -19 + 19 \beta_{1} - 19 \beta_{2} + 19 \beta_{3} ) q^{57} + ( 716 + 22 \beta_{3} ) q^{58} + ( 102 \beta_{1} - 153 \beta_{2} ) q^{59} + ( -20 \beta_{1} - 108 \beta_{2} ) q^{60} + ( -262 - 7 \beta_{1} + 269 \beta_{2} - 7 \beta_{3} ) q^{61} + ( -14 \beta_{1} - 108 \beta_{2} ) q^{62} + ( 312 - 104 \beta_{1} - 208 \beta_{2} - 104 \beta_{3} ) q^{63} + ( -509 + 51 \beta_{3} ) q^{64} + ( 390 - 3 \beta_{3} ) q^{65} + ( -39 - 15 \beta_{1} + 54 \beta_{2} - 15 \beta_{3} ) q^{66} + ( -197 - 162 \beta_{1} + 359 \beta_{2} - 162 \beta_{3} ) q^{67} + ( -906 + 258 \beta_{3} ) q^{68} + ( -8 + 17 \beta_{3} ) q^{69} + ( -1056 + 192 \beta_{1} + 864 \beta_{2} + 192 \beta_{3} ) q^{70} + ( 3 \beta_{1} - 783 \beta_{2} ) q^{71} + ( 546 - 78 \beta_{1} - 468 \beta_{2} - 78 \beta_{3} ) q^{72} + ( 88 \beta_{1} + 73 \beta_{2} ) q^{73} + ( -206 \beta_{1} + 648 \beta_{2} ) q^{74} + ( -7 - 19 \beta_{3} ) q^{75} + ( 342 + 19 \beta_{1} + 190 \beta_{2} - 190 \beta_{3} ) q^{76} + ( 36 + 36 \beta_{3} ) q^{77} + ( -48 \beta_{1} + 90 \beta_{2} ) q^{78} + ( -31 \beta_{1} + 181 \beta_{2} ) q^{79} + ( -104 + 104 \beta_{1} + 104 \beta_{3} ) q^{80} -649 \beta_{2} q^{81} + ( -25 + 61 \beta_{1} - 36 \beta_{2} + 61 \beta_{3} ) q^{82} + ( 649 + 161 \beta_{3} ) q^{83} + ( 204 - 52 \beta_{3} ) q^{84} + ( 858 - 237 \beta_{1} - 621 \beta_{2} - 237 \beta_{3} ) q^{85} + ( 544 - 166 \beta_{1} - 378 \beta_{2} - 166 \beta_{3} ) q^{86} + ( -22 - 41 \beta_{3} ) q^{87} + ( -153 - 9 \beta_{3} ) q^{88} + ( -340 + 79 \beta_{1} + 261 \beta_{2} + 79 \beta_{3} ) q^{89} + ( 260 \beta_{1} + 468 \beta_{2} ) q^{90} + ( -216 + 152 \beta_{1} + 64 \beta_{2} + 152 \beta_{3} ) q^{91} + ( 178 \beta_{1} + 216 \beta_{2} ) q^{92} + ( -6 \beta_{1} - 8 \beta_{2} ) q^{93} + ( -516 - 186 \beta_{3} ) q^{94} + ( -532 + 171 \beta_{1} + 171 \beta_{2} + 190 \beta_{3} ) q^{95} + ( 53 + 37 \beta_{3} ) q^{96} + ( 274 \beta_{1} + 25 \beta_{2} ) q^{97} + ( -89 \beta_{1} - 1440 \beta_{2} ) q^{98} + ( -390 - 78 \beta_{1} + 468 \beta_{2} - 78 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - q^{2} - 2q^{3} - 21q^{4} - 19q^{5} - q^{6} + 40q^{7} + 78q^{8} + 52q^{9} + O(q^{10}) \) \( 4q - q^{2} - 2q^{3} - 21q^{4} - 19q^{5} - q^{6} + 40q^{7} + 78q^{8} + 52q^{9} - 46q^{10} - 66q^{11} + 42q^{12} - 101q^{13} - 156q^{14} - 19q^{15} + 39q^{16} + 75q^{17} - 52q^{18} + 57q^{19} + 472q^{20} - 20q^{21} - 93q^{22} - q^{23} - 39q^{24} + 33q^{25} - 264q^{26} - 212q^{27} - 356q^{28} + 85q^{29} + 92q^{30} + 44q^{31} - 143q^{32} + 33q^{33} + 804q^{34} - 336q^{35} + 546q^{36} + 896q^{37} - 722q^{38} + 202q^{39} - 480q^{40} - 124q^{41} - 156q^{42} + 311q^{43} + 237q^{44} - 988q^{45} - 1240q^{46} - 411q^{47} + 39q^{48} + 196q^{49} + 1354q^{50} + 75q^{51} - 878q^{52} - 261q^{53} + 53q^{54} + 204q^{55} + 1656q^{56} - 57q^{57} + 2908q^{58} - 204q^{59} - 236q^{60} - 531q^{61} - 230q^{62} + 520q^{63} - 1934q^{64} + 1554q^{65} - 93q^{66} - 556q^{67} - 3108q^{68} + 2q^{69} - 1920q^{70} - 1563q^{71} + 1014q^{72} + 234q^{73} + 1090q^{74} - 66q^{75} + 1387q^{76} + 216q^{77} + 132q^{78} + 331q^{79} - 104q^{80} - 1298q^{81} + 11q^{82} + 2918q^{83} + 712q^{84} + 1479q^{85} + 922q^{86} - 170q^{87} - 630q^{88} - 601q^{89} + 1196q^{90} - 280q^{91} + 610q^{92} - 22q^{93} - 2436q^{94} - 1235q^{95} + 286q^{96} + 324q^{97} - 2969q^{98} - 858q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} + 19 x^{2} + 18 x + 324\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} + 19 \nu^{2} - 19 \nu + 324 \)\()/342\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} + 37 \)\()/19\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + 18 \beta_{2} + \beta_{1} - 19\)
\(\nu^{3}\)\(=\)\(19 \beta_{3} - 37\)

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
2.38600 + 4.13267i
−1.88600 3.26665i
2.38600 4.13267i
−1.88600 + 3.26665i
−2.38600 4.13267i −0.500000 0.866025i −7.38600 + 12.7929i −6.88600 11.9269i −2.38600 + 4.13267i 27.0880 32.3160 13.0000 22.5167i −32.8600 + 56.9152i
7.2 1.88600 + 3.26665i −0.500000 0.866025i −3.11400 + 5.39360i −2.61400 4.52758i 1.88600 3.26665i −7.08801 6.68399 13.0000 22.5167i 9.86001 17.0780i
11.1 −2.38600 + 4.13267i −0.500000 + 0.866025i −7.38600 12.7929i −6.88600 + 11.9269i −2.38600 4.13267i 27.0880 32.3160 13.0000 + 22.5167i −32.8600 56.9152i
11.2 1.88600 3.26665i −0.500000 + 0.866025i −3.11400 5.39360i −2.61400 + 4.52758i 1.88600 + 3.26665i −7.08801 6.68399 13.0000 + 22.5167i 9.86001 + 17.0780i
\(n\): 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 \( T_{2}^{4} + T_{2}^{3} + 19 T_{2}^{2} - 18 T_{2} + 324 \) acting on \(S_{4}^{\mathrm{new}}(19, [\chi])\).