Properties

Label 3.19.b.a
Level 3
Weight 19
Character orbit 3.b
Self dual Yes
Analytic conductor 6.162
Analytic rank 0
Dimension 1
CM disc. -3
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 3 \)
Weight: \( k \) = \( 19 \)
Character orbit: \([\chi]\) = 3.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: Yes
Analytic conductor: \(6.16158413129\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

\(f(q)\) \(=\) \( q - 19683q^{3} + 262144q^{4} + 77549186q^{7} + 387420489q^{9} + O(q^{10}) \) \( q - 19683q^{3} + 262144q^{4} + 77549186q^{7} + 387420489q^{9} - 5159780352q^{12} - 7197541846q^{13} + 68719476736q^{16} + 308559680858q^{19} - 1526400628038q^{21} + 3814697265625q^{25} - 7625597484987q^{27} + 20329053814784q^{28} - 50018992173358q^{31} + 101559956668416q^{36} - 23240947030054q^{37} + 141669216154818q^{39} - 730385642547286q^{43} - 1352605460594688q^{48} + 4385462651352147q^{49} - 1886792409677824q^{52} - 6073380198328014q^{57} - 9487161099916918q^{61} + 30044143561671954q^{63} + 18014398509481984q^{64} - 41747295001607494q^{67} - 29908998244279726q^{73} - 75084686279296875q^{75} + 80887068978839552q^{76} + 140655567501204338q^{79} + 150094635296999121q^{81} - 400136766236393472q^{84} - 558163511358237356q^{91} + 984523822948205514q^{93} + 140873967896062466q^{97} + O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3\mathbb{Z}\right)^\times\).

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
0
0 −19683.0 262144. 0 0 7.75492e7 0 3.87420e8 0
\(n\): 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
3.b Odd 1 CM by \(\Q(\sqrt{-3}) \) yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{19}^{\mathrm{new}}(3, [\chi])\).