Properties

Label 3.25.b.a
Level 3
Weight 25
Character orbit 3.b
Self dual Yes
Analytic conductor 10.949
Analytic rank 0
Dimension 1
CM disc. -3
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 3 \)
Weight: \( k \) = \( 25 \)
Character orbit: \([\chi]\) = 3.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: Yes
Analytic conductor: \(10.9490145677\)
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

\(f(q)\) \(=\) \( q + 531441q^{3} + 16777216q^{4} - 4119710398q^{7} + 282429536481q^{9} + O(q^{10}) \) \( q + 531441q^{3} + 16777216q^{4} - 4119710398q^{7} + 282429536481q^{9} + 8916100448256q^{12} + 41718377201762q^{13} + 281474976710656q^{16} - 4106686428542878q^{19} - 2189383013623518q^{21} + 59604644775390625q^{25} + 150094635296999121q^{27} - 69117271204691968q^{28} - 1294226424013985278q^{31} + 4738381338321616896q^{36} - 5178171888377036638q^{37} + 22170856098481599042q^{39} - 79863619813179602398q^{43} + 149587343098087735296q^{48} - 174609217617177095997q^{49} + 699918225483436654592q^{52} - 2182461542271255627198q^{57} + 176759412487640367842q^{61} - 1163527898143096029438q^{63} + 4722366482869645213696q^{64} + 6445029200733583780322q^{67} - 34892253008746830365758q^{73} + 31676352024078369140625q^{75} - 68898765255932429467648q^{76} + 79223687172395497630082q^{79} + 79766443076872509863361q^{81} - 36731751726292704165888q^{84} - 171867632345785055321276q^{91} - 687804985004416350125598q^{93} + 1377078975482507968769282q^{97} + O(q^{100}) \)

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 531441. 1.67772e7 0 0 −4.11971e9 0 2.82430e11 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 \(S_{25}^{\mathrm{new}}(3, [\chi])\).