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 disc. -3 Inner twists 2

Related objects

Newspace parameters

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

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

$q$-expansion

 $$f(q)$$ $$=$$ $$q$$ $$\mathstrut +\mathstrut 729q^{3}$$ $$\mathstrut +\mathstrut 4096q^{4}$$ $$\mathstrut -\mathstrut 153502q^{7}$$ $$\mathstrut +\mathstrut 531441q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$q$$ $$\mathstrut +\mathstrut 729q^{3}$$ $$\mathstrut +\mathstrut 4096q^{4}$$ $$\mathstrut -\mathstrut 153502q^{7}$$ $$\mathstrut +\mathstrut 531441q^{9}$$ $$\mathstrut +\mathstrut 2985984q^{12}$$ $$\mathstrut -\mathstrut 9397582q^{13}$$ $$\mathstrut +\mathstrut 16777216q^{16}$$ $$\mathstrut +\mathstrut 17886962q^{19}$$ $$\mathstrut -\mathstrut 111902958q^{21}$$ $$\mathstrut +\mathstrut 244140625q^{25}$$ $$\mathstrut +\mathstrut 387420489q^{27}$$ $$\mathstrut -\mathstrut 628744192q^{28}$$ $$\mathstrut -\mathstrut 530187838q^{31}$$ $$\mathstrut +\mathstrut 2176782336q^{36}$$ $$\mathstrut +\mathstrut 2826257618q^{37}$$ $$\mathstrut -\mathstrut 6850837278q^{39}$$ $$\mathstrut -\mathstrut 235885102q^{43}$$ $$\mathstrut +\mathstrut 12230590464q^{48}$$ $$\mathstrut +\mathstrut 9721576803q^{49}$$ $$\mathstrut -\mathstrut 38492495872q^{52}$$ $$\mathstrut +\mathstrut 13039595298q^{57}$$ $$\mathstrut +\mathstrut 74063873522q^{61}$$ $$\mathstrut -\mathstrut 81577256382q^{63}$$ $$\mathstrut +\mathstrut 68719476736q^{64}$$ $$\mathstrut -\mathstrut 151031344462q^{67}$$ $$\mathstrut +\mathstrut 104459767778q^{73}$$ $$\mathstrut +\mathstrut 177978515625q^{75}$$ $$\mathstrut +\mathstrut 73264996352q^{76}$$ $$\mathstrut -\mathstrut 444304748158q^{79}$$ $$\mathstrut +\mathstrut 282429536481q^{81}$$ $$\mathstrut -\mathstrut 458354515968q^{84}$$ $$\mathstrut +\mathstrut 1442547632164q^{91}$$ $$\mathstrut -\mathstrut 386506933902q^{93}$$ $$\mathstrut -\mathstrut 1662757858942q^{97}$$ $$\mathstrut +\mathstrut O(q^{100})$$

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
0 729.000 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. 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 $$S_{13}^{\mathrm{new}}(3, [\chi])$$.