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: Complex embeddings Normalized embeddings Satake parameters Satake angles

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 $$S_{25}^{\mathrm{new}}(3, [\chi])$$.