Properties

Label 19.6.a.a
Level 19
Weight 6
Character orbit 19.a
Self dual yes
Analytic conductor 3.047
Analytic rank 0
Dimension 1
CM no
Inner twists 1

Related objects

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Newspace parameters

Level: \( N \) = \( 19 \)
Weight: \( k \) = \( 6 \)
Character orbit: \([\chi]\) = 19.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.04729257645\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 6q^{2} + 4q^{3} + 4q^{4} + 54q^{5} - 24q^{6} + 248q^{7} + 168q^{8} - 227q^{9} + O(q^{10}) \) \( q - 6q^{2} + 4q^{3} + 4q^{4} + 54q^{5} - 24q^{6} + 248q^{7} + 168q^{8} - 227q^{9} - 324q^{10} + 204q^{11} + 16q^{12} - 370q^{13} - 1488q^{14} + 216q^{15} - 1136q^{16} + 1554q^{17} + 1362q^{18} + 361q^{19} + 216q^{20} + 992q^{21} - 1224q^{22} - 408q^{23} + 672q^{24} - 209q^{25} + 2220q^{26} - 1880q^{27} + 992q^{28} + 6174q^{29} - 1296q^{30} - 7840q^{31} + 1440q^{32} + 816q^{33} - 9324q^{34} + 13392q^{35} - 908q^{36} - 5146q^{37} - 2166q^{38} - 1480q^{39} + 9072q^{40} - 7830q^{41} - 5952q^{42} - 12532q^{43} + 816q^{44} - 12258q^{45} + 2448q^{46} + 2592q^{47} - 4544q^{48} + 44697q^{49} + 1254q^{50} + 6216q^{51} - 1480q^{52} - 20778q^{53} + 11280q^{54} + 11016q^{55} + 41664q^{56} + 1444q^{57} - 37044q^{58} + 18972q^{59} + 864q^{60} - 18418q^{61} + 47040q^{62} - 56296q^{63} + 27712q^{64} - 19980q^{65} - 4896q^{66} - 11548q^{67} + 6216q^{68} - 1632q^{69} - 80352q^{70} - 72984q^{71} - 38136q^{72} + 59114q^{73} + 30876q^{74} - 836q^{75} + 1444q^{76} + 50592q^{77} + 8880q^{78} - 44752q^{79} - 61344q^{80} + 47641q^{81} + 46980q^{82} - 27660q^{83} + 3968q^{84} + 83916q^{85} + 75192q^{86} + 24696q^{87} + 34272q^{88} + 20730q^{89} + 73548q^{90} - 91760q^{91} - 1632q^{92} - 31360q^{93} - 15552q^{94} + 19494q^{95} + 5760q^{96} + 14018q^{97} - 268182q^{98} - 46308q^{99} + O(q^{100}) \)

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
0
−6.00000 4.00000 4.00000 54.0000 −24.0000 248.000 168.000 −227.000 −324.000
\(n\): 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 19.6.a.a 1
3.b odd 2 1 171.6.a.d 1
4.b odd 2 1 304.6.a.a 1
5.b even 2 1 475.6.a.b 1
7.b odd 2 1 931.6.a.a 1
19.b odd 2 1 361.6.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.6.a.a 1 1.a even 1 1 trivial
171.6.a.d 1 3.b odd 2 1
304.6.a.a 1 4.b odd 2 1
361.6.a.c 1 19.b odd 2 1
475.6.a.b 1 5.b even 2 1
931.6.a.a 1 7.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 \( T_{2} + 6 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(19))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 6 T + 32 T^{2} \)
$3$ \( 1 - 4 T + 243 T^{2} \)
$5$ \( 1 - 54 T + 3125 T^{2} \)
$7$ \( 1 - 248 T + 16807 T^{2} \)
$11$ \( 1 - 204 T + 161051 T^{2} \)
$13$ \( 1 + 370 T + 371293 T^{2} \)
$17$ \( 1 - 1554 T + 1419857 T^{2} \)
$19$ \( 1 - 361 T \)
$23$ \( 1 + 408 T + 6436343 T^{2} \)
$29$ \( 1 - 6174 T + 20511149 T^{2} \)
$31$ \( 1 + 7840 T + 28629151 T^{2} \)
$37$ \( 1 + 5146 T + 69343957 T^{2} \)
$41$ \( 1 + 7830 T + 115856201 T^{2} \)
$43$ \( 1 + 12532 T + 147008443 T^{2} \)
$47$ \( 1 - 2592 T + 229345007 T^{2} \)
$53$ \( 1 + 20778 T + 418195493 T^{2} \)
$59$ \( 1 - 18972 T + 714924299 T^{2} \)
$61$ \( 1 + 18418 T + 844596301 T^{2} \)
$67$ \( 1 + 11548 T + 1350125107 T^{2} \)
$71$ \( 1 + 72984 T + 1804229351 T^{2} \)
$73$ \( 1 - 59114 T + 2073071593 T^{2} \)
$79$ \( 1 + 44752 T + 3077056399 T^{2} \)
$83$ \( 1 + 27660 T + 3939040643 T^{2} \)
$89$ \( 1 - 20730 T + 5584059449 T^{2} \)
$97$ \( 1 - 14018 T + 8587340257 T^{2} \)
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