# Properties

 Label 3.13.b.a Level $3$ Weight $13$ Character orbit 3.b Self dual yes Analytic conductor $2.742$ Analytic rank $0$ Dimension $1$ CM discriminant -3 Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3$$ Weight: $$k$$ $$=$$ $$13$$ Character orbit: $$[\chi]$$ $$=$$ 3.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$2.74198145183$$ 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

 $$f(q)$$ $$=$$ $$q + 729 q^{3} + 4096 q^{4} - 153502 q^{7} + 531441 q^{9}+O(q^{10})$$ q + 729 * q^3 + 4096 * q^4 - 153502 * q^7 + 531441 * q^9 $$q + 729 q^{3} + 4096 q^{4} - 153502 q^{7} + 531441 q^{9} + 2985984 q^{12} - 9397582 q^{13} + 16777216 q^{16} + 17886962 q^{19} - 111902958 q^{21} + 244140625 q^{25} + 387420489 q^{27} - 628744192 q^{28} - 530187838 q^{31} + 2176782336 q^{36} + 2826257618 q^{37} - 6850837278 q^{39} - 235885102 q^{43} + 12230590464 q^{48} + 9721576803 q^{49} - 38492495872 q^{52} + 13039595298 q^{57} + 74063873522 q^{61} - 81577256382 q^{63} + 68719476736 q^{64} - 151031344462 q^{67} + 104459767778 q^{73} + 177978515625 q^{75} + 73264996352 q^{76} - 444304748158 q^{79} + 282429536481 q^{81} - 458354515968 q^{84} + 1442547632164 q^{91} - 386506933902 q^{93} - 1662757858942 q^{97}+O(q^{100})$$ q + 729 * q^3 + 4096 * q^4 - 153502 * q^7 + 531441 * q^9 + 2985984 * q^12 - 9397582 * q^13 + 16777216 * q^16 + 17886962 * q^19 - 111902958 * q^21 + 244140625 * q^25 + 387420489 * q^27 - 628744192 * q^28 - 530187838 * q^31 + 2176782336 * q^36 + 2826257618 * q^37 - 6850837278 * q^39 - 235885102 * q^43 + 12230590464 * q^48 + 9721576803 * q^49 - 38492495872 * q^52 + 13039595298 * q^57 + 74063873522 * q^61 - 81577256382 * q^63 + 68719476736 * q^64 - 151031344462 * q^67 + 104459767778 * q^73 + 177978515625 * q^75 + 73264996352 * q^76 - 444304748158 * q^79 + 282429536481 * q^81 - 458354515968 * q^84 + 1442547632164 * q^91 - 386506933902 * q^93 - 1662757858942 * 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
0 729.000 4096.00 0 0 −153502. 0 531441. 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 3.13.b.a 1
3.b odd 2 1 CM 3.13.b.a 1
4.b odd 2 1 48.13.e.a 1
5.b even 2 1 75.13.c.a 1
5.c odd 4 2 75.13.d.a 2
8.b even 2 1 192.13.e.a 1
8.d odd 2 1 192.13.e.b 1
9.c even 3 2 81.13.d.a 2
9.d odd 6 2 81.13.d.a 2
12.b even 2 1 48.13.e.a 1
15.d odd 2 1 75.13.c.a 1
15.e even 4 2 75.13.d.a 2
24.f even 2 1 192.13.e.b 1
24.h odd 2 1 192.13.e.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.13.b.a 1 1.a even 1 1 trivial
3.13.b.a 1 3.b odd 2 1 CM
48.13.e.a 1 4.b odd 2 1
48.13.e.a 1 12.b even 2 1
75.13.c.a 1 5.b even 2 1
75.13.c.a 1 15.d odd 2 1
75.13.d.a 2 5.c odd 4 2
75.13.d.a 2 15.e even 4 2
81.13.d.a 2 9.c even 3 2
81.13.d.a 2 9.d odd 6 2
192.13.e.a 1 8.b even 2 1
192.13.e.a 1 24.h odd 2 1
192.13.e.b 1 8.d odd 2 1
192.13.e.b 1 24.f even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}$$ acting on $$S_{13}^{\mathrm{new}}(3, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 729$$
$5$ $$T$$
$7$ $$T + 153502$$
$11$ $$T$$
$13$ $$T + 9397582$$
$17$ $$T$$
$19$ $$T - 17886962$$
$23$ $$T$$
$29$ $$T$$
$31$ $$T + 530187838$$
$37$ $$T - 2826257618$$
$41$ $$T$$
$43$ $$T + 235885102$$
$47$ $$T$$
$53$ $$T$$
$59$ $$T$$
$61$ $$T - 74063873522$$
$67$ $$T + 151031344462$$
$71$ $$T$$
$73$ $$T - 104459767778$$
$79$ $$T + 444304748158$$
$83$ $$T$$
$89$ $$T$$
$97$ $$T + 1662757858942$$