# Properties

 Label 306.2.b.b Level $306$ Weight $2$ Character orbit 306.b Analytic conductor $2.443$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$306 = 2 \cdot 3^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 306.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.44342230185$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ Defining polynomial: $$x^{2} + 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{-2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} + \beta q^{5} -3 \beta q^{7} - q^{8} +O(q^{10})$$ $$q - q^{2} + q^{4} + \beta q^{5} -3 \beta q^{7} - q^{8} -\beta q^{10} -2 \beta q^{11} + 2 q^{13} + 3 \beta q^{14} + q^{16} + ( 3 + 2 \beta ) q^{17} + 2 q^{19} + \beta q^{20} + 2 \beta q^{22} -5 \beta q^{23} + 3 q^{25} -2 q^{26} -3 \beta q^{28} -5 \beta q^{29} + 3 \beta q^{31} - q^{32} + ( -3 - 2 \beta ) q^{34} + 6 q^{35} -3 \beta q^{37} -2 q^{38} -\beta q^{40} + 4 \beta q^{41} -4 q^{43} -2 \beta q^{44} + 5 \beta q^{46} + 12 q^{47} -11 q^{49} -3 q^{50} + 2 q^{52} -6 q^{53} + 4 q^{55} + 3 \beta q^{56} + 5 \beta q^{58} -6 q^{59} + 9 \beta q^{61} -3 \beta q^{62} + q^{64} + 2 \beta q^{65} -10 q^{67} + ( 3 + 2 \beta ) q^{68} -6 q^{70} + \beta q^{71} + 3 \beta q^{74} + 2 q^{76} -12 q^{77} + 9 \beta q^{79} + \beta q^{80} -4 \beta q^{82} + 6 q^{83} + ( -4 + 3 \beta ) q^{85} + 4 q^{86} + 2 \beta q^{88} -6 q^{89} -6 \beta q^{91} -5 \beta q^{92} -12 q^{94} + 2 \beta q^{95} -6 \beta q^{97} + 11 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} + O(q^{10})$$ $$2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} + 4 q^{13} + 2 q^{16} + 6 q^{17} + 4 q^{19} + 6 q^{25} - 4 q^{26} - 2 q^{32} - 6 q^{34} + 12 q^{35} - 4 q^{38} - 8 q^{43} + 24 q^{47} - 22 q^{49} - 6 q^{50} + 4 q^{52} - 12 q^{53} + 8 q^{55} - 12 q^{59} + 2 q^{64} - 20 q^{67} + 6 q^{68} - 12 q^{70} + 4 q^{76} - 24 q^{77} + 12 q^{83} - 8 q^{85} + 8 q^{86} - 12 q^{89} - 24 q^{94} + 22 q^{98} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/306\mathbb{Z}\right)^\times$$.

 $$n$$ $$37$$ $$137$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
271.1
 − 1.41421i 1.41421i
−1.00000 0 1.00000 1.41421i 0 4.24264i −1.00000 0 1.41421i
271.2 −1.00000 0 1.00000 1.41421i 0 4.24264i −1.00000 0 1.41421i
 $$n$$: 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
17.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 306.2.b.b 2
3.b odd 2 1 306.2.b.c yes 2
4.b odd 2 1 2448.2.c.k 2
12.b even 2 1 2448.2.c.i 2
17.b even 2 1 inner 306.2.b.b 2
17.c even 4 2 5202.2.a.bd 2
51.c odd 2 1 306.2.b.c yes 2
51.f odd 4 2 5202.2.a.t 2
68.d odd 2 1 2448.2.c.k 2
204.h even 2 1 2448.2.c.i 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
306.2.b.b 2 1.a even 1 1 trivial
306.2.b.b 2 17.b even 2 1 inner
306.2.b.c yes 2 3.b odd 2 1
306.2.b.c yes 2 51.c odd 2 1
2448.2.c.i 2 12.b even 2 1
2448.2.c.i 2 204.h even 2 1
2448.2.c.k 2 4.b odd 2 1
2448.2.c.k 2 68.d odd 2 1
5202.2.a.t 2 51.f odd 4 2
5202.2.a.bd 2 17.c even 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(306, [\chi])$$:

 $$T_{5}^{2} + 2$$ $$T_{47} - 12$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{2}$$
$3$ $$T^{2}$$
$5$ $$2 + T^{2}$$
$7$ $$18 + T^{2}$$
$11$ $$8 + T^{2}$$
$13$ $$( -2 + T )^{2}$$
$17$ $$17 - 6 T + T^{2}$$
$19$ $$( -2 + T )^{2}$$
$23$ $$50 + T^{2}$$
$29$ $$50 + T^{2}$$
$31$ $$18 + T^{2}$$
$37$ $$18 + T^{2}$$
$41$ $$32 + T^{2}$$
$43$ $$( 4 + T )^{2}$$
$47$ $$( -12 + T )^{2}$$
$53$ $$( 6 + T )^{2}$$
$59$ $$( 6 + T )^{2}$$
$61$ $$162 + T^{2}$$
$67$ $$( 10 + T )^{2}$$
$71$ $$2 + T^{2}$$
$73$ $$T^{2}$$
$79$ $$162 + T^{2}$$
$83$ $$( -6 + T )^{2}$$
$89$ $$( 6 + T )^{2}$$
$97$ $$72 + T^{2}$$