# 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 discriminant -3 Inner twists $2$

# Related objects

## Newspace parameters

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

## 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: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 531441 q^{3} + 16777216 q^{4} - 4119710398 q^{7} + 282429536481 q^{9}+O(q^{10})$$ q + 531441 * q^3 + 16777216 * q^4 - 4119710398 * q^7 + 282429536481 * q^9 $$q + 531441 q^{3} + 16777216 q^{4} - 4119710398 q^{7} + 282429536481 q^{9} + 8916100448256 q^{12} + 41718377201762 q^{13} + 281474976710656 q^{16} - 41\!\cdots\!78 q^{19}+ \cdots + 13\!\cdots\!82 q^{97}+O(q^{100})$$ q + 531441 * q^3 + 16777216 * q^4 - 4119710398 * q^7 + 282429536481 * q^9 + 8916100448256 * q^12 + 41718377201762 * q^13 + 281474976710656 * q^16 - 4106686428542878 * q^19 - 2189383013623518 * q^21 + 59604644775390625 * q^25 + 150094635296999121 * q^27 - 69117271204691968 * q^28 - 1294226424013985278 * q^31 + 4738381338321616896 * q^36 - 5178171888377036638 * q^37 + 22170856098481599042 * q^39 - 79863619813179602398 * q^43 + 149587343098087735296 * q^48 - 174609217617177095997 * q^49 + 699918225483436654592 * q^52 - 2182461542271255627198 * q^57 + 176759412487640367842 * q^61 - 1163527898143096029438 * q^63 + 4722366482869645213696 * q^64 + 6445029200733583780322 * q^67 - 34892253008746830365758 * q^73 + 31676352024078369140625 * q^75 - 68898765255932429467648 * q^76 + 79223687172395497630082 * q^79 + 79766443076872509863361 * q^81 - 36731751726292704165888 * q^84 - 171867632345785055321276 * q^91 - 687804985004416350125598 * q^93 + 1377078975482507968769282 * q^97

## 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 Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3.25.b.a 1
3.b odd 2 1 CM 3.25.b.a 1
4.b odd 2 1 48.25.e.a 1
12.b even 2 1 48.25.e.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.25.b.a 1 1.a even 1 1 trivial
3.25.b.a 1 3.b odd 2 1 CM
48.25.e.a 1 4.b odd 2 1
48.25.e.a 1 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}$$ acting on $$S_{25}^{\mathrm{new}}(3, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 531441$$
$5$ $$T$$
$7$ $$T + 4119710398$$
$11$ $$T$$
$13$ $$T - 41718377201762$$
$17$ $$T$$
$19$ $$T + 4106686428542878$$
$23$ $$T$$
$29$ $$T$$
$31$ $$T + 12\!\cdots\!78$$
$37$ $$T + 51\!\cdots\!38$$
$41$ $$T$$
$43$ $$T + 79\!\cdots\!98$$
$47$ $$T$$
$53$ $$T$$
$59$ $$T$$
$61$ $$T - 17\!\cdots\!42$$
$67$ $$T - 64\!\cdots\!22$$
$71$ $$T$$
$73$ $$T + 34\!\cdots\!58$$
$79$ $$T - 79\!\cdots\!82$$
$83$ $$T$$
$89$ $$T$$
$97$ $$T - 13\!\cdots\!82$$