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Properties

Label 3.7.b.a
Level 3
Weight 7
Character orbit 3.b
Self dual Yes
Analytic conductor 0.690
Analytic rank 0
Dimension 1
CM disc. -3
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 3 \)
Weight: \( k \) = \( 7 \)
Character orbit: \([\chi]\) = 3.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: Yes
Analytic conductor: \(0.69016225086\)
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

\(f(q)\) \(=\) \( q - 27q^{3} + 64q^{4} - 286q^{7} + 729q^{9} + O(q^{10}) \) \( q - 27q^{3} + 64q^{4} - 286q^{7} + 729q^{9} - 1728q^{12} + 506q^{13} + 4096q^{16} - 10582q^{19} + 7722q^{21} + 15625q^{25} - 19683q^{27} - 18304q^{28} + 35282q^{31} + 46656q^{36} - 89206q^{37} - 13662q^{39} + 111386q^{43} - 110592q^{48} - 35853q^{49} + 32384q^{52} + 285714q^{57} - 420838q^{61} - 208494q^{63} + 262144q^{64} + 172874q^{67} + 638066q^{73} - 421875q^{75} - 677248q^{76} - 204622q^{79} + 531441q^{81} + 494208q^{84} - 144716q^{91} - 952614q^{93} - 56446q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

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 −27.0000 64.0000 0 0 −286.000 0 729.000 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

There are no other newforms in \(S_{7}^{\mathrm{new}}(3, [\chi])\).