# Properties

 Label 7245.2.a.l Level $7245$ Weight $2$ Character orbit 7245.a Self dual yes Analytic conductor $57.852$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$7245 = 3^{2} \cdot 5 \cdot 7 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7245.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$57.8516162644$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 2415) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{4} - q^{5} - q^{7} - 3 q^{8}+O(q^{10})$$ q + q^2 - q^4 - q^5 - q^7 - 3 * q^8 $$q + q^{2} - q^{4} - q^{5} - q^{7} - 3 q^{8} - q^{10} + 2 q^{13} - q^{14} - q^{16} + 6 q^{17} - 4 q^{19} + q^{20} + q^{23} + q^{25} + 2 q^{26} + q^{28} + 2 q^{29} - 4 q^{31} + 5 q^{32} + 6 q^{34} + q^{35} - 6 q^{37} - 4 q^{38} + 3 q^{40} - 2 q^{41} + 4 q^{43} + q^{46} + q^{49} + q^{50} - 2 q^{52} - 2 q^{53} + 3 q^{56} + 2 q^{58} + 8 q^{59} - 2 q^{61} - 4 q^{62} + 7 q^{64} - 2 q^{65} + 4 q^{67} - 6 q^{68} + q^{70} - 8 q^{71} - 2 q^{73} - 6 q^{74} + 4 q^{76} + 4 q^{79} + q^{80} - 2 q^{82} + 4 q^{83} - 6 q^{85} + 4 q^{86} + 6 q^{89} - 2 q^{91} - q^{92} + 4 q^{95} - 14 q^{97} + q^{98}+O(q^{100})$$ q + q^2 - q^4 - q^5 - q^7 - 3 * q^8 - q^10 + 2 * q^13 - q^14 - q^16 + 6 * q^17 - 4 * q^19 + q^20 + q^23 + q^25 + 2 * q^26 + q^28 + 2 * q^29 - 4 * q^31 + 5 * q^32 + 6 * q^34 + q^35 - 6 * q^37 - 4 * q^38 + 3 * q^40 - 2 * q^41 + 4 * q^43 + q^46 + q^49 + q^50 - 2 * q^52 - 2 * q^53 + 3 * q^56 + 2 * q^58 + 8 * q^59 - 2 * q^61 - 4 * q^62 + 7 * q^64 - 2 * q^65 + 4 * q^67 - 6 * q^68 + q^70 - 8 * q^71 - 2 * q^73 - 6 * q^74 + 4 * q^76 + 4 * q^79 + q^80 - 2 * q^82 + 4 * q^83 - 6 * q^85 + 4 * q^86 + 6 * q^89 - 2 * q^91 - q^92 + 4 * q^95 - 14 * q^97 + q^98

## 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
1.00000 0 −1.00000 −1.00000 0 −1.00000 −3.00000 0 −1.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$5$$ $$1$$
$$7$$ $$1$$
$$23$$ $$-1$$

## 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 7245.2.a.l 1
3.b odd 2 1 2415.2.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2415.2.a.c 1 3.b odd 2 1
7245.2.a.l 1 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(7245))$$:

 $$T_{2} - 1$$ T2 - 1 $$T_{11}$$ T11 $$T_{13} - 2$$ T13 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T$$
$5$ $$T + 1$$
$7$ $$T + 1$$
$11$ $$T$$
$13$ $$T - 2$$
$17$ $$T - 6$$
$19$ $$T + 4$$
$23$ $$T - 1$$
$29$ $$T - 2$$
$31$ $$T + 4$$
$37$ $$T + 6$$
$41$ $$T + 2$$
$43$ $$T - 4$$
$47$ $$T$$
$53$ $$T + 2$$
$59$ $$T - 8$$
$61$ $$T + 2$$
$67$ $$T - 4$$
$71$ $$T + 8$$
$73$ $$T + 2$$
$79$ $$T - 4$$
$83$ $$T - 4$$
$89$ $$T - 6$$
$97$ $$T + 14$$