Properties

Label 19.6.a.b
Level 19
Weight 6
Character orbit 19.a
Self dual yes
Analytic conductor 3.047
Analytic rank 1
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: \(1\)
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

\(f(q)\) \(=\) \( q - 2q^{2} - q^{3} - 28q^{4} - 24q^{5} + 2q^{6} - 167q^{7} + 120q^{8} - 242q^{9} + O(q^{10}) \) \( q - 2q^{2} - q^{3} - 28q^{4} - 24q^{5} + 2q^{6} - 167q^{7} + 120q^{8} - 242q^{9} + 48q^{10} + 262q^{11} + 28q^{12} + 749q^{13} + 334q^{14} + 24q^{15} + 656q^{16} - 1597q^{17} + 484q^{18} - 361q^{19} + 672q^{20} + 167q^{21} - 524q^{22} - 2011q^{23} - 120q^{24} - 2549q^{25} - 1498q^{26} + 485q^{27} + 4676q^{28} - 1055q^{29} - 48q^{30} - 1548q^{31} - 5152q^{32} - 262q^{33} + 3194q^{34} + 4008q^{35} + 6776q^{36} + 9378q^{37} + 722q^{38} - 749q^{39} - 2880q^{40} - 10248q^{41} - 334q^{42} + 10544q^{43} - 7336q^{44} + 5808q^{45} + 4022q^{46} - 6912q^{47} - 656q^{48} + 11082q^{49} + 5098q^{50} + 1597q^{51} - 20972q^{52} - 35291q^{53} - 970q^{54} - 6288q^{55} - 20040q^{56} + 361q^{57} + 2110q^{58} + 33655q^{59} - 672q^{60} - 26218q^{61} + 3096q^{62} + 40414q^{63} - 10688q^{64} - 17976q^{65} + 524q^{66} + 45083q^{67} + 44716q^{68} + 2011q^{69} - 8016q^{70} + 30942q^{71} - 29040q^{72} + 46969q^{73} - 18756q^{74} + 2549q^{75} + 10108q^{76} - 43754q^{77} + 1498q^{78} - 64430q^{79} - 15744q^{80} + 58321q^{81} + 20496q^{82} - 13986q^{83} - 4676q^{84} + 38328q^{85} - 21088q^{86} + 1055q^{87} + 31440q^{88} - 137700q^{89} - 11616q^{90} - 125083q^{91} + 56308q^{92} + 1548q^{93} + 13824q^{94} + 8664q^{95} + 5152q^{96} - 22162q^{97} - 22164q^{98} - 63404q^{99} + O(q^{100}) \)

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
−2.00000 −1.00000 −28.0000 −24.0000 2.00000 −167.000 120.000 −242.000 48.0000
\(n\): 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.b 1
3.b odd 2 1 171.6.a.b 1
4.b odd 2 1 304.6.a.b 1
5.b even 2 1 475.6.a.a 1
7.b odd 2 1 931.6.a.b 1
19.b odd 2 1 361.6.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.6.a.b 1 1.a even 1 1 trivial
171.6.a.b 1 3.b odd 2 1
304.6.a.b 1 4.b odd 2 1
361.6.a.b 1 19.b odd 2 1
475.6.a.a 1 5.b even 2 1
931.6.a.b 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 \( T_{2} + 2 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(19))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T + 32 T^{2} \)
$3$ \( 1 + T + 243 T^{2} \)
$5$ \( 1 + 24 T + 3125 T^{2} \)
$7$ \( 1 + 167 T + 16807 T^{2} \)
$11$ \( 1 - 262 T + 161051 T^{2} \)
$13$ \( 1 - 749 T + 371293 T^{2} \)
$17$ \( 1 + 1597 T + 1419857 T^{2} \)
$19$ \( 1 + 361 T \)
$23$ \( 1 + 2011 T + 6436343 T^{2} \)
$29$ \( 1 + 1055 T + 20511149 T^{2} \)
$31$ \( 1 + 1548 T + 28629151 T^{2} \)
$37$ \( 1 - 9378 T + 69343957 T^{2} \)
$41$ \( 1 + 10248 T + 115856201 T^{2} \)
$43$ \( 1 - 10544 T + 147008443 T^{2} \)
$47$ \( 1 + 6912 T + 229345007 T^{2} \)
$53$ \( 1 + 35291 T + 418195493 T^{2} \)
$59$ \( 1 - 33655 T + 714924299 T^{2} \)
$61$ \( 1 + 26218 T + 844596301 T^{2} \)
$67$ \( 1 - 45083 T + 1350125107 T^{2} \)
$71$ \( 1 - 30942 T + 1804229351 T^{2} \)
$73$ \( 1 - 46969 T + 2073071593 T^{2} \)
$79$ \( 1 + 64430 T + 3077056399 T^{2} \)
$83$ \( 1 + 13986 T + 3939040643 T^{2} \)
$89$ \( 1 + 137700 T + 5584059449 T^{2} \)
$97$ \( 1 + 22162 T + 8587340257 T^{2} \)
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