# Properties

 Label 276.2.a.b Level $276$ Weight $2$ Character orbit 276.a Self dual yes Analytic conductor $2.204$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$276 = 2^{2} \cdot 3 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 276.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$2.20387109579$$ 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 Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + q^{3} + ( 2 + \beta ) q^{5} + \beta q^{7} + q^{9} +O(q^{10})$$ $$q + q^{3} + ( 2 + \beta ) q^{5} + \beta q^{7} + q^{9} -4 \beta q^{11} -4 \beta q^{13} + ( 2 + \beta ) q^{15} + ( 2 + 3 \beta ) q^{17} + ( -4 + 3 \beta ) q^{19} + \beta q^{21} + q^{23} + ( 1 + 4 \beta ) q^{25} + q^{27} + ( 6 - 2 \beta ) q^{29} + ( -4 - 2 \beta ) q^{31} -4 \beta q^{33} + ( 2 + 2 \beta ) q^{35} + ( -6 - 2 \beta ) q^{37} -4 \beta q^{39} -2 q^{41} + ( -4 + \beta ) q^{43} + ( 2 + \beta ) q^{45} + ( -2 + 6 \beta ) q^{47} -5 q^{49} + ( 2 + 3 \beta ) q^{51} + ( 6 + \beta ) q^{53} + ( -8 - 8 \beta ) q^{55} + ( -4 + 3 \beta ) q^{57} + ( -6 + 2 \beta ) q^{59} + ( 6 + 2 \beta ) q^{61} + \beta q^{63} + ( -8 - 8 \beta ) q^{65} + ( -12 - 3 \beta ) q^{67} + q^{69} -8 \beta q^{71} + ( 6 + 4 \beta ) q^{73} + ( 1 + 4 \beta ) q^{75} -8 q^{77} + 5 \beta q^{79} + q^{81} + ( -4 + 4 \beta ) q^{83} + ( 10 + 8 \beta ) q^{85} + ( 6 - 2 \beta ) q^{87} + ( 6 - 5 \beta ) q^{89} -8 q^{91} + ( -4 - 2 \beta ) q^{93} + ( -2 + 2 \beta ) q^{95} + ( 2 - 10 \beta ) q^{97} -4 \beta q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{3} + 4q^{5} + 2q^{9} + O(q^{10})$$ $$2q + 2q^{3} + 4q^{5} + 2q^{9} + 4q^{15} + 4q^{17} - 8q^{19} + 2q^{23} + 2q^{25} + 2q^{27} + 12q^{29} - 8q^{31} + 4q^{35} - 12q^{37} - 4q^{41} - 8q^{43} + 4q^{45} - 4q^{47} - 10q^{49} + 4q^{51} + 12q^{53} - 16q^{55} - 8q^{57} - 12q^{59} + 12q^{61} - 16q^{65} - 24q^{67} + 2q^{69} + 12q^{73} + 2q^{75} - 16q^{77} + 2q^{81} - 8q^{83} + 20q^{85} + 12q^{87} + 12q^{89} - 16q^{91} - 8q^{93} - 4q^{95} + 4q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −1.41421 1.41421
0 1.00000 0 0.585786 0 −1.41421 0 1.00000 0
1.2 0 1.00000 0 3.41421 0 1.41421 0 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$23$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 276.2.a.b 2
3.b odd 2 1 828.2.a.e 2
4.b odd 2 1 1104.2.a.l 2
5.b even 2 1 6900.2.a.m 2
5.c odd 4 2 6900.2.f.l 4
8.b even 2 1 4416.2.a.bc 2
8.d odd 2 1 4416.2.a.bi 2
12.b even 2 1 3312.2.a.s 2
23.b odd 2 1 6348.2.a.h 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
276.2.a.b 2 1.a even 1 1 trivial
828.2.a.e 2 3.b odd 2 1
1104.2.a.l 2 4.b odd 2 1
3312.2.a.s 2 12.b even 2 1
4416.2.a.bc 2 8.b even 2 1
4416.2.a.bi 2 8.d odd 2 1
6348.2.a.h 2 23.b odd 2 1
6900.2.a.m 2 5.b even 2 1
6900.2.f.l 4 5.c odd 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - 4 T_{5} + 2$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(276))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$( -1 + T )^{2}$$
$5$ $$2 - 4 T + T^{2}$$
$7$ $$-2 + T^{2}$$
$11$ $$-32 + T^{2}$$
$13$ $$-32 + T^{2}$$
$17$ $$-14 - 4 T + T^{2}$$
$19$ $$-2 + 8 T + T^{2}$$
$23$ $$( -1 + T )^{2}$$
$29$ $$28 - 12 T + T^{2}$$
$31$ $$8 + 8 T + T^{2}$$
$37$ $$28 + 12 T + T^{2}$$
$41$ $$( 2 + T )^{2}$$
$43$ $$14 + 8 T + T^{2}$$
$47$ $$-68 + 4 T + T^{2}$$
$53$ $$34 - 12 T + T^{2}$$
$59$ $$28 + 12 T + T^{2}$$
$61$ $$28 - 12 T + T^{2}$$
$67$ $$126 + 24 T + T^{2}$$
$71$ $$-128 + T^{2}$$
$73$ $$4 - 12 T + T^{2}$$
$79$ $$-50 + T^{2}$$
$83$ $$-16 + 8 T + T^{2}$$
$89$ $$-14 - 12 T + T^{2}$$
$97$ $$-196 - 4 T + T^{2}$$