# Properties

 Label 7448.2.a.bc Level $7448$ Weight $2$ Character orbit 7448.a Self dual yes Analytic conductor $59.473$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$7448 = 2^{3} \cdot 7^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7448.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$59.4725794254$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1064) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{5})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta ) q^{3} + ( -1 + 2 \beta ) q^{5} + ( -1 + 3 \beta ) q^{9} +O(q^{10})$$ $$q + ( 1 + \beta ) q^{3} + ( -1 + 2 \beta ) q^{5} + ( -1 + 3 \beta ) q^{9} + ( -2 + 3 \beta ) q^{11} + ( 5 - 2 \beta ) q^{13} + ( 1 + 3 \beta ) q^{15} + ( 4 - 3 \beta ) q^{17} - q^{19} + ( 3 + 2 \beta ) q^{23} + ( -1 + 2 \beta ) q^{27} + ( -6 + \beta ) q^{29} + ( 4 - \beta ) q^{31} + ( 1 + 4 \beta ) q^{33} + ( -5 + 4 \beta ) q^{37} + ( 3 + \beta ) q^{39} + ( -4 + 9 \beta ) q^{41} + ( 6 - 8 \beta ) q^{43} + ( 7 + \beta ) q^{45} -7 q^{47} + ( 1 - 2 \beta ) q^{51} + ( 3 - 3 \beta ) q^{53} + ( 8 - \beta ) q^{55} + ( -1 - \beta ) q^{57} + ( 1 + 6 \beta ) q^{59} + ( 3 + 6 \beta ) q^{61} + ( -9 + 8 \beta ) q^{65} + ( -4 - \beta ) q^{67} + ( 5 + 7 \beta ) q^{69} + ( -9 + 8 \beta ) q^{71} + ( 5 - 3 \beta ) q^{73} + ( 2 - 4 \beta ) q^{79} + ( 4 - 6 \beta ) q^{81} + ( 13 - \beta ) q^{83} + ( -10 + 5 \beta ) q^{85} + ( -5 - 4 \beta ) q^{87} + ( -4 + 2 \beta ) q^{89} + ( 3 + 2 \beta ) q^{93} + ( 1 - 2 \beta ) q^{95} + ( 7 - 2 \beta ) q^{97} + 11 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 3 q^{3} + q^{9} + O(q^{10})$$ $$2 q + 3 q^{3} + q^{9} - q^{11} + 8 q^{13} + 5 q^{15} + 5 q^{17} - 2 q^{19} + 8 q^{23} - 11 q^{29} + 7 q^{31} + 6 q^{33} - 6 q^{37} + 7 q^{39} + q^{41} + 4 q^{43} + 15 q^{45} - 14 q^{47} + 3 q^{53} + 15 q^{55} - 3 q^{57} + 8 q^{59} + 12 q^{61} - 10 q^{65} - 9 q^{67} + 17 q^{69} - 10 q^{71} + 7 q^{73} + 2 q^{81} + 25 q^{83} - 15 q^{85} - 14 q^{87} - 6 q^{89} + 8 q^{93} + 12 q^{97} + 22 q^{99} + O(q^{100})$$

## 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
 −0.618034 1.61803
0 0.381966 0 −2.23607 0 0 0 −2.85410 0
1.2 0 2.61803 0 2.23607 0 0 0 3.85410 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
$$2$$ $$1$$
$$7$$ $$-1$$
$$19$$ $$1$$

## 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 7448.2.a.bc 2
7.b odd 2 1 1064.2.a.a 2
21.c even 2 1 9576.2.a.bm 2
28.d even 2 1 2128.2.a.n 2
56.e even 2 1 8512.2.a.i 2
56.h odd 2 1 8512.2.a.bf 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1064.2.a.a 2 7.b odd 2 1
2128.2.a.n 2 28.d even 2 1
7448.2.a.bc 2 1.a even 1 1 trivial
8512.2.a.i 2 56.e even 2 1
8512.2.a.bf 2 56.h odd 2 1
9576.2.a.bm 2 21.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(7448))$$:

 $$T_{3}^{2} - 3 T_{3} + 1$$ $$T_{5}^{2} - 5$$ $$T_{11}^{2} + T_{11} - 11$$ $$T_{13}^{2} - 8 T_{13} + 11$$ $$T_{17}^{2} - 5 T_{17} - 5$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$1 - 3 T + T^{2}$$
$5$ $$-5 + T^{2}$$
$7$ $$T^{2}$$
$11$ $$-11 + T + T^{2}$$
$13$ $$11 - 8 T + T^{2}$$
$17$ $$-5 - 5 T + T^{2}$$
$19$ $$( 1 + T )^{2}$$
$23$ $$11 - 8 T + T^{2}$$
$29$ $$29 + 11 T + T^{2}$$
$31$ $$11 - 7 T + T^{2}$$
$37$ $$-11 + 6 T + T^{2}$$
$41$ $$-101 - T + T^{2}$$
$43$ $$-76 - 4 T + T^{2}$$
$47$ $$( 7 + T )^{2}$$
$53$ $$-9 - 3 T + T^{2}$$
$59$ $$-29 - 8 T + T^{2}$$
$61$ $$-9 - 12 T + T^{2}$$
$67$ $$19 + 9 T + T^{2}$$
$71$ $$-55 + 10 T + T^{2}$$
$73$ $$1 - 7 T + T^{2}$$
$79$ $$-20 + T^{2}$$
$83$ $$155 - 25 T + T^{2}$$
$89$ $$4 + 6 T + T^{2}$$
$97$ $$31 - 12 T + T^{2}$$