# Properties

 Label 1225.4.a.l Level $1225$ Weight $4$ Character orbit 1225.a Self dual yes Analytic conductor $72.277$ Analytic rank $1$ Dimension $1$ CM discriminant -7 Inner twists $2$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$1225 = 5^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1225.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$72.2773397570$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 49) Fricke sign: $$-1$$ Sato-Tate group: $N(\mathrm{U}(1))$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 5q^{2} + 17q^{4} + 45q^{8} - 27q^{9} + O(q^{10})$$ $$q + 5q^{2} + 17q^{4} + 45q^{8} - 27q^{9} - 68q^{11} + 89q^{16} - 135q^{18} - 340q^{22} + 40q^{23} - 166q^{29} + 85q^{32} - 459q^{36} - 450q^{37} + 180q^{43} - 1156q^{44} + 200q^{46} - 590q^{53} - 830q^{58} - 287q^{64} + 740q^{67} + 688q^{71} - 1215q^{72} - 2250q^{74} - 1384q^{79} + 729q^{81} + 900q^{86} - 3060q^{88} + 680q^{92} + 1836q^{99} + O(q^{100})$$

## 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
5.00000 0 17.0000 0 0 0 45.0000 −27.0000 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
$$5$$ $$1$$
$$7$$ $$-1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 CM by $$\Q(\sqrt{-7})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1225.4.a.l 1
5.b even 2 1 49.4.a.a 1
7.b odd 2 1 CM 1225.4.a.l 1
15.d odd 2 1 441.4.a.m 1
20.d odd 2 1 784.4.a.k 1
35.c odd 2 1 49.4.a.a 1
35.i odd 6 2 49.4.c.d 2
35.j even 6 2 49.4.c.d 2
105.g even 2 1 441.4.a.m 1
105.o odd 6 2 441.4.e.a 2
105.p even 6 2 441.4.e.a 2
140.c even 2 1 784.4.a.k 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.4.a.a 1 5.b even 2 1
49.4.a.a 1 35.c odd 2 1
49.4.c.d 2 35.i odd 6 2
49.4.c.d 2 35.j even 6 2
441.4.a.m 1 15.d odd 2 1
441.4.a.m 1 105.g even 2 1
441.4.e.a 2 105.o odd 6 2
441.4.e.a 2 105.p even 6 2
784.4.a.k 1 20.d odd 2 1
784.4.a.k 1 140.c even 2 1
1225.4.a.l 1 1.a even 1 1 trivial
1225.4.a.l 1 7.b odd 2 1 CM

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1225))$$:

 $$T_{2} - 5$$ $$T_{3}$$ $$T_{19}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-5 + T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T$$
$11$ $$68 + T$$
$13$ $$T$$
$17$ $$T$$
$19$ $$T$$
$23$ $$-40 + T$$
$29$ $$166 + T$$
$31$ $$T$$
$37$ $$450 + T$$
$41$ $$T$$
$43$ $$-180 + T$$
$47$ $$T$$
$53$ $$590 + T$$
$59$ $$T$$
$61$ $$T$$
$67$ $$-740 + T$$
$71$ $$-688 + T$$
$73$ $$T$$
$79$ $$1384 + T$$
$83$ $$T$$
$89$ $$T$$
$97$ $$T$$
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