Properties

Label 19.4.a.b
Level 19
Weight 4
Character orbit 19.a
Self dual yes
Analytic conductor 1.121
Analytic rank 0
Dimension 3
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 19.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.12103629011\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.3144.1
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( 1 + \beta_{1} - \beta_{2} ) q^{2} + ( -\beta_{1} + 2 \beta_{2} ) q^{3} + ( 8 - \beta_{1} - 2 \beta_{2} ) q^{4} + ( 5 - 3 \beta_{1} + 2 \beta_{2} ) q^{5} + ( -24 + 3 \beta_{1} + 4 \beta_{2} ) q^{6} + ( -13 + 4 \beta_{1} ) q^{7} + ( 12 - \beta_{1} - 8 \beta_{2} ) q^{8} + ( 19 - 3 \beta_{1} - 6 \beta_{2} ) q^{9} +O(q^{10})\) \( q + ( 1 + \beta_{1} - \beta_{2} ) q^{2} + ( -\beta_{1} + 2 \beta_{2} ) q^{3} + ( 8 - \beta_{1} - 2 \beta_{2} ) q^{4} + ( 5 - 3 \beta_{1} + 2 \beta_{2} ) q^{5} + ( -24 + 3 \beta_{1} + 4 \beta_{2} ) q^{6} + ( -13 + 4 \beta_{1} ) q^{7} + ( 12 - \beta_{1} - 8 \beta_{2} ) q^{8} + ( 19 - 3 \beta_{1} - 6 \beta_{2} ) q^{9} + ( -31 + 10 \beta_{1} - 5 \beta_{2} ) q^{10} + ( 5 + 3 \beta_{1} - 2 \beta_{2} ) q^{11} + ( -42 - 15 \beta_{1} + 26 \beta_{2} ) q^{12} + ( 22 + 5 \beta_{1} - 6 \beta_{2} ) q^{13} + ( 11 - 17 \beta_{1} + 21 \beta_{2} ) q^{14} + ( 50 - 18 \beta_{1} + 8 \beta_{2} ) q^{15} + ( 14 + 13 \beta_{1} - 22 \beta_{2} ) q^{16} + ( 1 + 22 \beta_{1} + 4 \beta_{2} ) q^{17} + ( 55 + 16 \beta_{1} - 43 \beta_{2} ) q^{18} -19 q^{19} + ( 34 - 22 \beta_{1} + 20 \beta_{2} ) q^{20} + ( -8 + 33 \beta_{1} - 34 \beta_{2} ) q^{21} + ( 41 - 5 \beta_{2} ) q^{22} + ( -42 + 11 \beta_{1} + 14 \beta_{2} ) q^{23} + ( -174 - 25 \beta_{1} + 58 \beta_{2} ) q^{24} + ( -6 - 49 \beta_{1} + 30 \beta_{2} ) q^{25} + ( 106 + 11 \beta_{1} - 30 \beta_{2} ) q^{26} + ( -126 - 13 \beta_{1} + 14 \beta_{2} ) q^{27} + ( -176 + 17 \beta_{1} + 18 \beta_{2} ) q^{28} + ( 144 - 25 \beta_{1} - 30 \beta_{2} ) q^{29} + ( -130 + 76 \beta_{1} - 62 \beta_{2} ) q^{30} + ( -32 - 56 \beta_{1} + 12 \beta_{2} ) q^{31} + ( 194 - 13 \beta_{1} + 10 \beta_{2} ) q^{32} + ( -50 + 8 \beta_{1} + 12 \beta_{2} ) q^{33} + ( 97 - 17 \beta_{1} + 55 \beta_{2} ) q^{34} + ( -153 + 71 \beta_{1} - 50 \beta_{2} ) q^{35} + ( 386 + 20 \beta_{1} - 104 \beta_{2} ) q^{36} + ( -122 + 32 \beta_{1} + 44 \beta_{2} ) q^{37} + ( -19 - 19 \beta_{1} + 19 \beta_{2} ) q^{38} + ( -142 - 3 \beta_{1} + 58 \beta_{2} ) q^{39} + ( -30 - 4 \beta_{1} + 22 \beta_{2} ) q^{40} + ( 314 + 6 \beta_{1} + 8 \beta_{2} ) q^{41} + ( 496 - 75 \beta_{1} - 28 \beta_{2} ) q^{42} + ( -163 + 29 \beta_{1} - 110 \beta_{2} ) q^{43} + ( 46 + 12 \beta_{1} - 40 \beta_{2} ) q^{44} + ( 77 - 51 \beta_{1} + 50 \beta_{2} ) q^{45} + ( -102 - 39 \beta_{1} + 106 \beta_{2} ) q^{46} + ( 39 - 33 \beta_{1} - 18 \beta_{2} ) q^{47} + ( -510 + 29 \beta_{1} + 90 \beta_{2} ) q^{48} + ( -14 - 88 \beta_{1} + 32 \beta_{2} ) q^{49} + ( -570 + 73 \beta_{1} - 2 \beta_{2} ) q^{50} + ( 44 + 113 \beta_{1} - 58 \beta_{2} ) q^{51} + ( 266 + 25 \beta_{1} - 126 \beta_{2} ) q^{52} + ( 266 + 29 \beta_{1} - 10 \beta_{2} ) q^{53} + ( -330 - 99 \beta_{1} + 142 \beta_{2} ) q^{54} + ( -69 + 19 \beta_{1} - 10 \beta_{2} ) q^{55} + ( -324 - 39 \beta_{1} + 96 \beta_{2} ) q^{56} + ( 19 \beta_{1} - 38 \beta_{2} ) q^{57} + ( 264 + 139 \beta_{1} - 284 \beta_{2} ) q^{58} + ( 128 - 53 \beta_{1} - 66 \beta_{2} ) q^{59} + ( 484 - 124 \beta_{1} + 32 \beta_{2} ) q^{60} + ( 285 + 23 \beta_{1} + 110 \beta_{2} ) q^{61} + ( -476 + 36 \beta_{1} - 44 \beta_{2} ) q^{62} + ( -463 + 31 \beta_{1} + 54 \beta_{2} ) q^{63} + ( -86 + 113 \beta_{1} - 14 \beta_{2} ) q^{64} + ( -84 + 4 \beta_{1} + 8 \beta_{2} ) q^{65} + ( -110 - 46 \beta_{1} + 102 \beta_{2} ) q^{66} + ( -94 + 29 \beta_{1} + 46 \beta_{2} ) q^{67} + ( -508 - 7 \beta_{1} + 2 \beta_{2} ) q^{68} + ( 286 + 111 \beta_{1} - 162 \beta_{2} ) q^{69} + ( 723 - 274 \beta_{1} + 145 \beta_{2} ) q^{70} + ( 270 - 28 \beta_{1} + 64 \beta_{2} ) q^{71} + ( 1002 + 134 \beta_{1} - 314 \beta_{2} ) q^{72} + ( 265 - 176 \beta_{1} + 8 \beta_{2} ) q^{73} + ( -326 - 110 \beta_{1} + 318 \beta_{2} ) q^{74} + ( 758 - 209 \beta_{1} - 34 \beta_{2} ) q^{75} + ( -152 + 19 \beta_{1} + 38 \beta_{2} ) q^{76} + ( 23 - 31 \beta_{1} + 50 \beta_{2} ) q^{77} + ( -682 - 81 \beta_{1} + 310 \beta_{2} ) q^{78} + ( 56 + 206 \beta_{1} + 8 \beta_{2} ) q^{79} + ( -524 + 172 \beta_{1} - 72 \beta_{2} ) q^{80} + ( -179 + 156 \beta_{1} - 120 \beta_{2} ) q^{81} + ( 278 + 316 \beta_{1} - 278 \beta_{2} ) q^{82} + ( -232 - 130 \beta_{1} + 60 \beta_{2} ) q^{83} + ( 362 + 279 \beta_{1} - 458 \beta_{2} ) q^{84} + ( -423 + 153 \beta_{1} - 126 \beta_{2} ) q^{85} + ( 1001 - 302 \beta_{1} - 109 \beta_{2} ) q^{86} + ( -610 - 299 \beta_{1} + 458 \beta_{2} ) q^{87} + ( 150 - 6 \beta_{1} - 102 \beta_{2} ) q^{88} + ( -40 + 168 \beta_{1} - 220 \beta_{2} ) q^{89} + ( -679 + 178 \beta_{1} - 29 \beta_{2} ) q^{90} + ( -182 - 29 \beta_{1} + 118 \beta_{2} ) q^{91} + ( -954 - 45 \beta_{1} + 230 \beta_{2} ) q^{92} + ( 376 - 236 \beta_{1} ) q^{93} + ( 3 + 54 \beta_{1} - 159 \beta_{2} ) q^{94} + ( -95 + 57 \beta_{1} - 38 \beta_{2} ) q^{95} + ( 246 - 249 \beta_{1} + 374 \beta_{2} ) q^{96} + ( -852 - 90 \beta_{1} + 196 \beta_{2} ) q^{97} + ( -830 + 106 \beta_{1} - 66 \beta_{2} ) q^{98} + ( 113 + 21 \beta_{1} - 110 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{2} + q^{3} + 21q^{4} + 14q^{5} - 65q^{6} - 35q^{7} + 27q^{8} + 48q^{9} + O(q^{10}) \) \( 3q + 3q^{2} + q^{3} + 21q^{4} + 14q^{5} - 65q^{6} - 35q^{7} + 27q^{8} + 48q^{9} - 88q^{10} + 16q^{11} - 115q^{12} + 65q^{13} + 37q^{14} + 140q^{15} + 33q^{16} + 29q^{17} + 138q^{18} - 57q^{19} + 100q^{20} - 25q^{21} + 118q^{22} - 101q^{23} - 489q^{24} - 37q^{25} + 299q^{26} - 377q^{27} - 493q^{28} + 377q^{29} - 376q^{30} - 140q^{31} + 579q^{32} - 130q^{33} + 329q^{34} - 438q^{35} + 1074q^{36} - 290q^{37} - 57q^{38} - 371q^{39} - 72q^{40} + 956q^{41} + 1385q^{42} - 570q^{43} + 110q^{44} + 230q^{45} - 239q^{46} + 66q^{47} - 1411q^{48} - 98q^{49} - 1639q^{50} + 187q^{51} + 697q^{52} + 817q^{53} - 947q^{54} - 198q^{55} - 915q^{56} - 19q^{57} + 647q^{58} + 265q^{59} + 1360q^{60} + 988q^{61} - 1436q^{62} - 1304q^{63} - 159q^{64} - 240q^{65} - 274q^{66} - 207q^{67} - 1529q^{68} + 807q^{69} + 2040q^{70} + 846q^{71} + 2826q^{72} + 627q^{73} - 770q^{74} + 2031q^{75} - 399q^{76} + 88q^{77} - 1817q^{78} + 382q^{79} - 1472q^{80} - 501q^{81} + 872q^{82} - 766q^{83} + 907q^{84} - 1242q^{85} + 2592q^{86} - 1671q^{87} + 342q^{88} - 172q^{89} - 1888q^{90} - 457q^{91} - 2677q^{92} + 892q^{93} - 96q^{94} - 266q^{95} + 863q^{96} - 2450q^{97} - 2450q^{98} + 250q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 16 x - 8\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{2} - \nu - 10 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(2 \beta_{2} + \beta_{1} + 10\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−3.20905
4.73549
−0.526440
−3.96257 6.71610 7.70200 18.1342 −26.6130 −25.8362 1.18085 18.1060 −71.8581
1.2 1.89080 2.95388 −4.42486 −1.51710 5.58521 5.94196 −23.4930 −18.2746 −2.86853
1.3 5.07177 −8.66998 17.7229 −2.61710 −43.9722 −15.1058 49.3121 48.1686 −13.2733
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 19.4.a.b 3
3.b odd 2 1 171.4.a.f 3
4.b odd 2 1 304.4.a.i 3
5.b even 2 1 475.4.a.f 3
5.c odd 4 2 475.4.b.f 6
7.b odd 2 1 931.4.a.c 3
8.b even 2 1 1216.4.a.s 3
8.d odd 2 1 1216.4.a.u 3
11.b odd 2 1 2299.4.a.h 3
19.b odd 2 1 361.4.a.i 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.4.a.b 3 1.a even 1 1 trivial
171.4.a.f 3 3.b odd 2 1
304.4.a.i 3 4.b odd 2 1
361.4.a.i 3 19.b odd 2 1
475.4.a.f 3 5.b even 2 1
475.4.b.f 6 5.c odd 4 2
931.4.a.c 3 7.b odd 2 1
1216.4.a.s 3 8.b even 2 1
1216.4.a.u 3 8.d odd 2 1
2299.4.a.h 3 11.b odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(19\) \(1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} - 3 T_{2}^{2} - 18 T_{2} + 38 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(19))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 3 T + 6 T^{2} - 10 T^{3} + 48 T^{4} - 192 T^{5} + 512 T^{6} \)
$3$ \( 1 - T + 17 T^{2} + 118 T^{3} + 459 T^{4} - 729 T^{5} + 19683 T^{6} \)
$5$ \( 1 - 14 T + 304 T^{2} - 3572 T^{3} + 38000 T^{4} - 218750 T^{5} + 1953125 T^{6} \)
$7$ \( 1 + 35 T + 1176 T^{2} + 21691 T^{3} + 403368 T^{4} + 4117715 T^{5} + 40353607 T^{6} \)
$11$ \( 1 - 16 T + 3942 T^{2} - 41410 T^{3} + 5246802 T^{4} - 28344976 T^{5} + 2357947691 T^{6} \)
$13$ \( 1 - 65 T + 7335 T^{2} - 280762 T^{3} + 16114995 T^{4} - 313742585 T^{5} + 10604499373 T^{6} \)
$17$ \( 1 - 29 T + 5514 T^{2} - 503573 T^{3} + 27090282 T^{4} - 699989501 T^{5} + 118587876497 T^{6} \)
$19$ \( ( 1 + 19 T )^{3} \)
$23$ \( 1 + 101 T + 31877 T^{2} + 2079558 T^{3} + 387847459 T^{4} + 14951624789 T^{5} + 1801152661463 T^{6} \)
$29$ \( 1 - 377 T + 81935 T^{2} - 13844910 T^{3} + 1998312715 T^{4} - 224248392017 T^{5} + 14507145975869 T^{6} \)
$31$ \( 1 + 140 T + 51757 T^{2} + 5897128 T^{3} + 1541892787 T^{4} + 124250515340 T^{5} + 26439622160671 T^{6} \)
$37$ \( 1 + 290 T + 105187 T^{2} + 19377292 T^{3} + 5328037111 T^{4} + 744060658610 T^{5} + 129961739795077 T^{6} \)
$41$ \( 1 - 956 T + 508879 T^{2} - 163355096 T^{3} + 35072449559 T^{4} - 4541099654396 T^{5} + 327381934393961 T^{6} \)
$43$ \( 1 + 570 T + 145938 T^{2} + 24674476 T^{3} + 11603092566 T^{4} + 3603176937930 T^{5} + 502592611936843 T^{6} \)
$47$ \( 1 - 66 T + 280158 T^{2} - 10764012 T^{3} + 29086844034 T^{4} - 711428211714 T^{5} + 1119130473102767 T^{6} \)
$53$ \( 1 - 817 T + 657711 T^{2} - 260089834 T^{3} + 97918040547 T^{4} - 18108283042393 T^{5} + 3299763591802133 T^{6} \)
$59$ \( 1 - 265 T + 458145 T^{2} - 77293258 T^{3} + 94093361955 T^{4} - 11177841414865 T^{5} + 8662995818654939 T^{6} \)
$61$ \( 1 - 988 T + 726644 T^{2} - 371638582 T^{3} + 164934381764 T^{4} - 50902129868668 T^{5} + 11694146092834141 T^{6} \)
$67$ \( 1 + 207 T + 842361 T^{2} + 117000634 T^{3} + 253351021443 T^{4} + 18724885108983 T^{5} + 27206534396294947 T^{6} \)
$71$ \( 1 - 846 T + 1246593 T^{2} - 603857484 T^{3} + 446169347223 T^{4} - 108372840197166 T^{5} + 45848500718449031 T^{6} \)
$73$ \( 1 - 627 T + 811566 T^{2} - 342245479 T^{3} + 315712970622 T^{4} - 94886559883203 T^{5} + 58871586708267913 T^{6} \)
$79$ \( 1 - 382 T + 809229 T^{2} - 432705284 T^{3} + 398981456931 T^{4} - 92859408009022 T^{5} + 119851595982618319 T^{6} \)
$83$ \( 1 + 766 T + 1679713 T^{2} + 797249332 T^{3} + 960438057131 T^{4} + 250436326000654 T^{5} + 186940255267540403 T^{6} \)
$89$ \( 1 + 172 T + 1270123 T^{2} + 165585880 T^{3} + 895397341187 T^{4} + 85480782045292 T^{5} + 350356403707485209 T^{6} \)
$97$ \( 1 + 2450 T + 4122563 T^{2} + 4668536612 T^{3} + 3762551940899 T^{4} + 2040781412076050 T^{5} + 760231058654565217 T^{6} \)
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