Properties

Label 19.5.b.b
Level $19$
Weight $5$
Character orbit 19.b
Analytic conductor $1.964$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 19 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 19.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.96402929859\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.0.12107488.1
Defining polynomial: \(x^{4} + 35 x^{2} + 142\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} -\beta_{3} q^{3} + ( -3 + 3 \beta_{2} ) q^{4} + ( -10 - \beta_{2} ) q^{5} + ( 3 + 7 \beta_{2} ) q^{6} + ( 31 + 6 \beta_{2} ) q^{7} + ( 4 \beta_{1} + 3 \beta_{3} ) q^{8} + ( -54 - 31 \beta_{2} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} -\beta_{3} q^{3} + ( -3 + 3 \beta_{2} ) q^{4} + ( -10 - \beta_{2} ) q^{5} + ( 3 + 7 \beta_{2} ) q^{6} + ( 31 + 6 \beta_{2} ) q^{7} + ( 4 \beta_{1} + 3 \beta_{3} ) q^{8} + ( -54 - 31 \beta_{2} ) q^{9} + ( -7 \beta_{1} - \beta_{3} ) q^{10} + ( 64 - 17 \beta_{2} ) q^{11} + ( -18 \beta_{1} - 9 \beta_{3} ) q^{12} + ( -10 \beta_{1} + 11 \beta_{3} ) q^{13} + ( 13 \beta_{1} + 6 \beta_{3} ) q^{14} + ( 6 \beta_{1} + 14 \beta_{3} ) q^{15} + ( -133 + 39 \beta_{2} ) q^{16} + ( 63 + 24 \beta_{2} ) q^{17} + ( 39 \beta_{1} - 31 \beta_{3} ) q^{18} + ( -38 \beta_{1} + 57 \beta_{2} + 19 \beta_{3} ) q^{19} + ( -24 - 30 \beta_{2} ) q^{20} + ( -36 \beta_{1} - 55 \beta_{3} ) q^{21} + ( 115 \beta_{1} - 17 \beta_{3} ) q^{22} + ( 37 - 113 \beta_{2} ) q^{23} + ( 417 + 121 \beta_{2} ) q^{24} + ( -507 + 21 \beta_{2} ) q^{25} + ( 157 - 107 \beta_{2} ) q^{26} + ( 186 \beta_{1} + 97 \beta_{3} ) q^{27} + ( 231 + 93 \beta_{2} ) q^{28} + ( -208 \beta_{1} - 3 \beta_{3} ) q^{29} + ( -156 - 80 \beta_{2} ) q^{30} + ( -242 \beta_{1} - 56 \beta_{3} ) q^{31} + ( -186 \beta_{1} + 87 \beta_{3} ) q^{32} + ( 102 \beta_{1} + 4 \beta_{3} ) q^{33} + ( -9 \beta_{1} + 24 \beta_{3} ) q^{34} + ( -418 - 97 \beta_{2} ) q^{35} + ( -1512 - 162 \beta_{2} ) q^{36} + ( 282 \beta_{1} - 72 \beta_{3} ) q^{37} + ( 665 - 171 \beta_{1} - 247 \beta_{2} + 57 \beta_{3} ) q^{38} + ( 1455 + 271 \beta_{2} ) q^{39} + ( -46 \beta_{1} - 46 \beta_{3} ) q^{40} + ( 430 \beta_{1} - 36 \beta_{3} ) q^{41} + ( 849 + 277 \beta_{2} ) q^{42} + ( 922 - 351 \beta_{2} ) q^{43} + ( -1110 + 192 \beta_{2} ) q^{44} + ( 1098 + 395 \beta_{2} ) q^{45} + ( 376 \beta_{1} - 113 \beta_{3} ) q^{46} + ( -1852 - 85 \beta_{2} ) q^{47} + ( -234 \beta_{1} - 23 \beta_{3} ) q^{48} + ( -792 + 408 \beta_{2} ) q^{49} + ( -570 \beta_{1} + 21 \beta_{3} ) q^{50} + ( -144 \beta_{1} - 159 \beta_{3} ) q^{51} + ( 318 \beta_{1} + 69 \beta_{3} ) q^{52} + ( 204 \beta_{1} + 63 \beta_{3} ) q^{53} + ( -3825 - 121 \beta_{2} ) q^{54} + ( -334 + 123 \beta_{2} ) q^{55} + ( 160 \beta_{1} + 189 \beta_{3} ) q^{56} + ( 2451 - 342 \beta_{1} + 323 \beta_{2} - 228 \beta_{3} ) q^{57} + ( 3961 - 603 \beta_{2} ) q^{58} + ( -250 \beta_{1} + 303 \beta_{3} ) q^{59} + ( 180 \beta_{1} + 144 \beta_{3} ) q^{60} + ( -46 + 171 \beta_{2} ) q^{61} + ( 4766 - 334 \beta_{2} ) q^{62} + ( -5022 - 1471 \beta_{2} ) q^{63} + ( 1145 - 543 \beta_{2} ) q^{64} + ( 4 \beta_{1} - 144 \beta_{3} ) q^{65} + ( -1950 + 278 \beta_{2} ) q^{66} + ( -428 \beta_{1} + 151 \beta_{3} ) q^{67} + ( 1107 + 189 \beta_{2} ) q^{68} + ( 678 \beta_{1} + 415 \beta_{3} ) q^{69} + ( -127 \beta_{1} - 97 \beta_{3} ) q^{70} + ( -516 \beta_{1} - 390 \beta_{3} ) q^{71} + ( -402 \beta_{1} - 658 \beta_{3} ) q^{72} + ( -3963 + 342 \beta_{2} ) q^{73} + ( -5142 + 1350 \beta_{2} ) q^{74} + ( -126 \beta_{1} + 423 \beta_{3} ) q^{75} + ( 3078 + 798 \beta_{1} + 57 \beta_{3} ) q^{76} + ( 148 - 245 \beta_{2} ) q^{77} + ( 642 \beta_{1} + 271 \beta_{3} ) q^{78} + ( -566 \beta_{1} + 76 \beta_{3} ) q^{79} + ( 628 - 296 \beta_{2} ) q^{80} + ( 9279 + 1798 \beta_{2} ) q^{81} + ( -8062 + 1542 \beta_{2} ) q^{82} + ( 8188 + 262 \beta_{2} ) q^{83} + ( -558 \beta_{1} - 603 \beta_{3} ) q^{84} + ( -1062 - 327 \beta_{2} ) q^{85} + ( 1975 \beta_{1} - 351 \beta_{3} ) q^{86} + ( -1029 - 1549 \beta_{2} ) q^{87} + ( 154 \beta_{1} - 80 \beta_{3} ) q^{88} + ( 1374 \beta_{1} - 726 \beta_{3} ) q^{89} + ( -87 \beta_{1} + 395 \beta_{3} ) q^{90} + ( 266 \beta_{1} + 545 \beta_{3} ) q^{91} + ( -6213 + 111 \beta_{2} ) q^{92} + ( -8286 - 3430 \beta_{2} ) q^{93} + ( -1597 \beta_{1} - 85 \beta_{3} ) q^{94} + ( -1026 + 152 \beta_{1} - 627 \beta_{2} - 228 \beta_{3} ) q^{95} + ( 11187 + 1395 \beta_{2} ) q^{96} + ( -640 \beta_{1} - 70 \beta_{3} ) q^{97} + ( -2016 \beta_{1} + 408 \beta_{3} ) q^{98} + ( 6030 - 539 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 6q^{4} - 42q^{5} + 26q^{6} + 136q^{7} - 278q^{9} + O(q^{10}) \) \( 4q - 6q^{4} - 42q^{5} + 26q^{6} + 136q^{7} - 278q^{9} + 222q^{11} - 454q^{16} + 300q^{17} + 114q^{19} - 156q^{20} - 78q^{23} + 1910q^{24} - 1986q^{25} + 414q^{26} + 1110q^{28} - 784q^{30} - 1866q^{35} - 6372q^{36} + 2166q^{38} + 6362q^{39} + 3950q^{42} + 2986q^{43} - 4056q^{44} + 5182q^{45} - 7578q^{47} - 2352q^{49} - 15542q^{54} - 1090q^{55} + 10450q^{57} + 14638q^{58} + 158q^{61} + 18396q^{62} - 23030q^{63} + 3494q^{64} - 7244q^{66} + 4806q^{68} - 15168q^{73} - 17868q^{74} + 12312q^{76} + 102q^{77} + 1920q^{80} + 40712q^{81} - 29164q^{82} + 33276q^{83} - 4902q^{85} - 7214q^{87} - 24630q^{92} - 40004q^{93} - 5358q^{95} + 47538q^{96} + 23042q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 35 x^{2} + 142\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{2} + 19 \)\()/3\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} + 28 \nu \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(3 \beta_{2} - 19\)
\(\nu^{3}\)\(=\)\(3 \beta_{3} - 28 \beta_{1}\)

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
18.1
5.50600i
2.16425i
2.16425i
5.50600i
5.50600i 4.25064i −14.3160 −6.22800 −23.4040 8.36799 9.27207i 62.9321 34.2913i
18.2 2.16425i 16.8206i 11.3160 −14.7720 36.4040 59.6320 59.1188i −201.932 31.9704i
18.3 2.16425i 16.8206i 11.3160 −14.7720 36.4040 59.6320 59.1188i −201.932 31.9704i
18.4 5.50600i 4.25064i −14.3160 −6.22800 −23.4040 8.36799 9.27207i 62.9321 34.2913i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 19.5.b.b 4
3.b odd 2 1 171.5.c.c 4
4.b odd 2 1 304.5.e.c 4
19.b odd 2 1 inner 19.5.b.b 4
57.d even 2 1 171.5.c.c 4
76.d even 2 1 304.5.e.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.5.b.b 4 1.a even 1 1 trivial
19.5.b.b 4 19.b odd 2 1 inner
171.5.c.c 4 3.b odd 2 1
171.5.c.c 4 57.d even 2 1
304.5.e.c 4 4.b odd 2 1
304.5.e.c 4 76.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 35 T_{2}^{2} + 142 \) acting on \(S_{5}^{\mathrm{new}}(19, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 142 + 35 T^{2} + T^{4} \)
$3$ \( 5112 + 301 T^{2} + T^{4} \)
$5$ \( ( 92 + 21 T + T^{2} )^{2} \)
$7$ \( ( 499 - 68 T + T^{2} )^{2} \)
$11$ \( ( -2194 - 111 T + T^{2} )^{2} \)
$13$ \( 276731872 + 37061 T^{2} + T^{4} \)
$17$ \( ( -4887 - 150 T + T^{2} )^{2} \)
$19$ \( 16983563041 - 14856594 T + 26714 T^{2} - 114 T^{3} + T^{4} \)
$23$ \( ( -232654 + 39 T + T^{2} )^{2} \)
$29$ \( 321440583928 + 1533173 T^{2} + T^{4} \)
$31$ \( 2573095457248 + 3346028 T^{2} + T^{4} \)
$37$ \( 1246928875488 + 3815820 T^{2} + T^{4} \)
$41$ \( 671445252832 + 6459116 T^{2} + T^{4} \)
$43$ \( ( -1691156 - 1493 T + T^{2} )^{2} \)
$47$ \( ( 3457274 + 3789 T + T^{2} )^{2} \)
$53$ \( 1649118527352 + 2985381 T^{2} + T^{4} \)
$59$ \( 147332683451872 + 27852509 T^{2} + T^{4} \)
$61$ \( ( -532088 - 79 T + T^{2} )^{2} \)
$67$ \( 23408787518008 + 11594213 T^{2} + T^{4} \)
$71$ \( 82524177119232 + 60333300 T^{2} + T^{4} \)
$73$ \( ( 12244671 + 7584 T + T^{2} )^{2} \)
$79$ \( 33729223648 + 11832620 T^{2} + T^{4} \)
$83$ \( ( 67953008 - 16638 T + T^{2} )^{2} \)
$89$ \( 9681864523757568 + 198789912 T^{2} + T^{4} \)
$97$ \( 68352898480000 + 16975700 T^{2} + T^{4} \)
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