# Properties

 Label 1035.2.a.k Level $1035$ Weight $2$ Character orbit 1035.a Self dual yes Analytic conductor $8.265$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1035 = 3^{2} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1035.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$8.26451660920$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{6})$$ Defining polynomial: $$x^{2} - 6$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 345) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + 4 q^{4} + q^{5} - q^{7} + 2 \beta q^{8} +O(q^{10})$$ $$q + \beta q^{2} + 4 q^{4} + q^{5} - q^{7} + 2 \beta q^{8} + \beta q^{10} -\beta q^{11} + ( 2 + \beta ) q^{13} -\beta q^{14} + 4 q^{16} + ( 3 + \beta ) q^{17} + ( 2 + \beta ) q^{19} + 4 q^{20} -6 q^{22} + q^{23} + q^{25} + ( 6 + 2 \beta ) q^{26} -4 q^{28} + ( -3 - 3 \beta ) q^{29} + ( 5 - 2 \beta ) q^{31} + ( 6 + 3 \beta ) q^{34} - q^{35} + ( -1 + 2 \beta ) q^{37} + ( 6 + 2 \beta ) q^{38} + 2 \beta q^{40} + ( -3 - \beta ) q^{41} + 2 q^{43} -4 \beta q^{44} + \beta q^{46} + ( -6 - \beta ) q^{47} -6 q^{49} + \beta q^{50} + ( 8 + 4 \beta ) q^{52} + ( -3 + \beta ) q^{53} -\beta q^{55} -2 \beta q^{56} + ( -18 - 3 \beta ) q^{58} + ( -3 - 3 \beta ) q^{59} + ( 8 - 3 \beta ) q^{61} + ( -12 + 5 \beta ) q^{62} -8 q^{64} + ( 2 + \beta ) q^{65} -7 q^{67} + ( 12 + 4 \beta ) q^{68} -\beta q^{70} + ( -3 + 3 \beta ) q^{71} + ( 2 - 3 \beta ) q^{73} + ( 12 - \beta ) q^{74} + ( 8 + 4 \beta ) q^{76} + \beta q^{77} -4 q^{79} + 4 q^{80} + ( -6 - 3 \beta ) q^{82} + ( -3 - 5 \beta ) q^{83} + ( 3 + \beta ) q^{85} + 2 \beta q^{86} -12 q^{88} + ( 12 + 2 \beta ) q^{89} + ( -2 - \beta ) q^{91} + 4 q^{92} + ( -6 - 6 \beta ) q^{94} + ( 2 + \beta ) q^{95} + ( 8 - 2 \beta ) q^{97} -6 \beta q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 8q^{4} + 2q^{5} - 2q^{7} + O(q^{10})$$ $$2q + 8q^{4} + 2q^{5} - 2q^{7} + 4q^{13} + 8q^{16} + 6q^{17} + 4q^{19} + 8q^{20} - 12q^{22} + 2q^{23} + 2q^{25} + 12q^{26} - 8q^{28} - 6q^{29} + 10q^{31} + 12q^{34} - 2q^{35} - 2q^{37} + 12q^{38} - 6q^{41} + 4q^{43} - 12q^{47} - 12q^{49} + 16q^{52} - 6q^{53} - 36q^{58} - 6q^{59} + 16q^{61} - 24q^{62} - 16q^{64} + 4q^{65} - 14q^{67} + 24q^{68} - 6q^{71} + 4q^{73} + 24q^{74} + 16q^{76} - 8q^{79} + 8q^{80} - 12q^{82} - 6q^{83} + 6q^{85} - 24q^{88} + 24q^{89} - 4q^{91} + 8q^{92} - 12q^{94} + 4q^{95} + 16q^{97} + O(q^{100})$$

## 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
 −2.44949 2.44949
−2.44949 0 4.00000 1.00000 0 −1.00000 −4.89898 0 −2.44949
1.2 2.44949 0 4.00000 1.00000 0 −1.00000 4.89898 0 2.44949
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$5$$ $$-1$$
$$23$$ $$-1$$

## 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 1035.2.a.k 2
3.b odd 2 1 345.2.a.i 2
5.b even 2 1 5175.2.a.bl 2
12.b even 2 1 5520.2.a.bi 2
15.d odd 2 1 1725.2.a.y 2
15.e even 4 2 1725.2.b.m 4
69.c even 2 1 7935.2.a.t 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
345.2.a.i 2 3.b odd 2 1
1035.2.a.k 2 1.a even 1 1 trivial
1725.2.a.y 2 15.d odd 2 1
1725.2.b.m 4 15.e even 4 2
5175.2.a.bl 2 5.b even 2 1
5520.2.a.bi 2 12.b even 2 1
7935.2.a.t 2 69.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1035))$$:

 $$T_{2}^{2} - 6$$ $$T_{7} + 1$$ $$T_{11}^{2} - 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-6 + T^{2}$$
$3$ $$T^{2}$$
$5$ $$( -1 + T )^{2}$$
$7$ $$( 1 + T )^{2}$$
$11$ $$-6 + T^{2}$$
$13$ $$-2 - 4 T + T^{2}$$
$17$ $$3 - 6 T + T^{2}$$
$19$ $$-2 - 4 T + T^{2}$$
$23$ $$( -1 + T )^{2}$$
$29$ $$-45 + 6 T + T^{2}$$
$31$ $$1 - 10 T + T^{2}$$
$37$ $$-23 + 2 T + T^{2}$$
$41$ $$3 + 6 T + T^{2}$$
$43$ $$( -2 + T )^{2}$$
$47$ $$30 + 12 T + T^{2}$$
$53$ $$3 + 6 T + T^{2}$$
$59$ $$-45 + 6 T + T^{2}$$
$61$ $$10 - 16 T + T^{2}$$
$67$ $$( 7 + T )^{2}$$
$71$ $$-45 + 6 T + T^{2}$$
$73$ $$-50 - 4 T + T^{2}$$
$79$ $$( 4 + T )^{2}$$
$83$ $$-141 + 6 T + T^{2}$$
$89$ $$120 - 24 T + T^{2}$$
$97$ $$40 - 16 T + T^{2}$$