# Properties

 Label 1575.2.a.m Level $1575$ Weight $2$ Character orbit 1575.a Self dual yes Analytic conductor $12.576$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1575 = 3^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1575.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$12.5764383184$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 315) 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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta ) q^{2} + ( 1 - 2 \beta ) q^{4} + q^{7} + ( -3 + \beta ) q^{8} +O(q^{10})$$ $$q + ( -1 + \beta ) q^{2} + ( 1 - 2 \beta ) q^{4} + q^{7} + ( -3 + \beta ) q^{8} + ( 2 + 2 \beta ) q^{11} + ( 2 - 2 \beta ) q^{13} + ( -1 + \beta ) q^{14} + 3 q^{16} + ( -2 - 4 \beta ) q^{17} -2 \beta q^{19} + 2 q^{22} + ( -2 + 4 \beta ) q^{23} + ( -6 + 4 \beta ) q^{26} + ( 1 - 2 \beta ) q^{28} + 8 q^{29} + 6 \beta q^{31} + ( 3 + \beta ) q^{32} + ( -6 + 2 \beta ) q^{34} + 6 q^{37} + ( -4 + 2 \beta ) q^{38} + ( -2 - 4 \beta ) q^{41} + ( 4 - 4 \beta ) q^{43} + ( -6 - 2 \beta ) q^{44} + ( 10 - 6 \beta ) q^{46} + 4 q^{47} + q^{49} + ( 10 - 6 \beta ) q^{52} + ( -8 + 2 \beta ) q^{53} + ( -3 + \beta ) q^{56} + ( -8 + 8 \beta ) q^{58} -4 q^{59} + 6 q^{61} + ( 12 - 6 \beta ) q^{62} + ( -7 + 2 \beta ) q^{64} + ( 4 + 8 \beta ) q^{67} + 14 q^{68} + ( 2 + 6 \beta ) q^{71} + ( -2 + 10 \beta ) q^{73} + ( -6 + 6 \beta ) q^{74} + ( 8 - 2 \beta ) q^{76} + ( 2 + 2 \beta ) q^{77} + 4 \beta q^{79} + ( -6 + 2 \beta ) q^{82} + 8 q^{83} + ( -12 + 8 \beta ) q^{86} + ( -2 - 4 \beta ) q^{88} + ( 6 - 8 \beta ) q^{89} + ( 2 - 2 \beta ) q^{91} + ( -18 + 8 \beta ) q^{92} + ( -4 + 4 \beta ) q^{94} + ( 6 - 2 \beta ) q^{97} + ( -1 + \beta ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} + 2q^{4} + 2q^{7} - 6q^{8} + O(q^{10})$$ $$2q - 2q^{2} + 2q^{4} + 2q^{7} - 6q^{8} + 4q^{11} + 4q^{13} - 2q^{14} + 6q^{16} - 4q^{17} + 4q^{22} - 4q^{23} - 12q^{26} + 2q^{28} + 16q^{29} + 6q^{32} - 12q^{34} + 12q^{37} - 8q^{38} - 4q^{41} + 8q^{43} - 12q^{44} + 20q^{46} + 8q^{47} + 2q^{49} + 20q^{52} - 16q^{53} - 6q^{56} - 16q^{58} - 8q^{59} + 12q^{61} + 24q^{62} - 14q^{64} + 8q^{67} + 28q^{68} + 4q^{71} - 4q^{73} - 12q^{74} + 16q^{76} + 4q^{77} - 12q^{82} + 16q^{83} - 24q^{86} - 4q^{88} + 12q^{89} + 4q^{91} - 36q^{92} - 8q^{94} + 12q^{97} - 2q^{98} + O(q^{100})$$

## 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
−2.41421 0 3.82843 0 0 1.00000 −4.41421 0 0
1.2 0.414214 0 −1.82843 0 0 1.00000 −1.58579 0 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
$$3$$ $$1$$
$$5$$ $$1$$
$$7$$ $$-1$$

## 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 1575.2.a.m 2
3.b odd 2 1 1575.2.a.u 2
5.b even 2 1 315.2.a.f yes 2
5.c odd 4 2 1575.2.d.j 4
15.d odd 2 1 315.2.a.c 2
15.e even 4 2 1575.2.d.h 4
20.d odd 2 1 5040.2.a.bx 2
35.c odd 2 1 2205.2.a.y 2
60.h even 2 1 5040.2.a.bu 2
105.g even 2 1 2205.2.a.p 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.a.c 2 15.d odd 2 1
315.2.a.f yes 2 5.b even 2 1
1575.2.a.m 2 1.a even 1 1 trivial
1575.2.a.u 2 3.b odd 2 1
1575.2.d.h 4 15.e even 4 2
1575.2.d.j 4 5.c odd 4 2
2205.2.a.p 2 105.g even 2 1
2205.2.a.y 2 35.c odd 2 1
5040.2.a.bu 2 60.h even 2 1
5040.2.a.bx 2 20.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1575))$$:

 $$T_{2}^{2} + 2 T_{2} - 1$$ $$T_{11}^{2} - 4 T_{11} - 4$$ $$T_{13}^{2} - 4 T_{13} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 + 2 T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$( -1 + T )^{2}$$
$11$ $$-4 - 4 T + T^{2}$$
$13$ $$-4 - 4 T + T^{2}$$
$17$ $$-28 + 4 T + T^{2}$$
$19$ $$-8 + T^{2}$$
$23$ $$-28 + 4 T + T^{2}$$
$29$ $$( -8 + T )^{2}$$
$31$ $$-72 + T^{2}$$
$37$ $$( -6 + T )^{2}$$
$41$ $$-28 + 4 T + T^{2}$$
$43$ $$-16 - 8 T + T^{2}$$
$47$ $$( -4 + T )^{2}$$
$53$ $$56 + 16 T + T^{2}$$
$59$ $$( 4 + T )^{2}$$
$61$ $$( -6 + T )^{2}$$
$67$ $$-112 - 8 T + T^{2}$$
$71$ $$-68 - 4 T + T^{2}$$
$73$ $$-196 + 4 T + T^{2}$$
$79$ $$-32 + T^{2}$$
$83$ $$( -8 + T )^{2}$$
$89$ $$-92 - 12 T + T^{2}$$
$97$ $$28 - 12 T + T^{2}$$