# Properties

 Label 306.2.b.d 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: $$2$$ Twist minimal: no (minimal twist has level 34) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

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

## 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 2.82843i 0 0 1.00000 0 2.82843i
271.2 1.00000 0 1.00000 2.82843i 0 0 1.00000 0 2.82843i
 $$n$$: 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.d 2
3.b odd 2 1 34.2.b.a 2
4.b odd 2 1 2448.2.c.n 2
12.b even 2 1 272.2.b.a 2
15.d odd 2 1 850.2.b.f 2
15.e even 4 2 850.2.d.i 4
17.b even 2 1 inner 306.2.b.d 2
17.c even 4 2 5202.2.a.u 2
21.c even 2 1 1666.2.b.c 2
24.f even 2 1 1088.2.b.a 2
24.h odd 2 1 1088.2.b.b 2
51.c odd 2 1 34.2.b.a 2
51.f odd 4 2 578.2.a.d 2
51.g odd 8 2 578.2.c.a 2
51.g odd 8 2 578.2.c.d 2
51.i even 16 8 578.2.d.g 8
68.d odd 2 1 2448.2.c.n 2
204.h even 2 1 272.2.b.a 2
204.l even 4 2 4624.2.a.s 2
255.h odd 2 1 850.2.b.f 2
255.o even 4 2 850.2.d.i 4
357.c even 2 1 1666.2.b.c 2
408.b odd 2 1 1088.2.b.b 2
408.h even 2 1 1088.2.b.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
34.2.b.a 2 3.b odd 2 1
34.2.b.a 2 51.c odd 2 1
272.2.b.a 2 12.b even 2 1
272.2.b.a 2 204.h even 2 1
306.2.b.d 2 1.a even 1 1 trivial
306.2.b.d 2 17.b even 2 1 inner
578.2.a.d 2 51.f odd 4 2
578.2.c.a 2 51.g odd 8 2
578.2.c.d 2 51.g odd 8 2
578.2.d.g 8 51.i even 16 8
850.2.b.f 2 15.d odd 2 1
850.2.b.f 2 255.h odd 2 1
850.2.d.i 4 15.e even 4 2
850.2.d.i 4 255.o even 4 2
1088.2.b.a 2 24.f even 2 1
1088.2.b.a 2 408.h even 2 1
1088.2.b.b 2 24.h odd 2 1
1088.2.b.b 2 408.b odd 2 1
1666.2.b.c 2 21.c even 2 1
1666.2.b.c 2 357.c even 2 1
2448.2.c.n 2 4.b odd 2 1
2448.2.c.n 2 68.d odd 2 1
4624.2.a.s 2 204.l even 4 2
5202.2.a.u 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 $$S_{2}^{\mathrm{new}}(306, [\chi])$$:

 $$T_{5}^{2} + 8$$ $$T_{47}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{2}$$
$3$ $$T^{2}$$
$5$ $$8 + T^{2}$$
$7$ $$T^{2}$$
$11$ $$8 + T^{2}$$
$13$ $$( -2 + T )^{2}$$
$17$ $$17 - 6 T + T^{2}$$
$19$ $$( 4 + T )^{2}$$
$23$ $$32 + T^{2}$$
$29$ $$8 + T^{2}$$
$31$ $$T^{2}$$
$37$ $$72 + T^{2}$$
$41$ $$32 + T^{2}$$
$43$ $$( 4 + T )^{2}$$
$47$ $$T^{2}$$
$53$ $$( 6 + T )^{2}$$
$59$ $$( 12 + T )^{2}$$
$61$ $$72 + T^{2}$$
$67$ $$( 4 + T )^{2}$$
$71$ $$32 + T^{2}$$
$73$ $$T^{2}$$
$79$ $$288 + T^{2}$$
$83$ $$( -12 + T )^{2}$$
$89$ $$( 6 + T )^{2}$$
$97$ $$288 + T^{2}$$