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

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

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 \(S_{19}^{\mathrm{new}}(3, [\chi])\).