# Properties

 Label 1225.2.a.e Level $1225$ Weight $2$ Character orbit 1225.a Self dual yes Analytic conductor $9.782$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

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## Newspace parameters

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

## Newform invariants

 Self dual: yes Analytic conductor: $$9.78167424761$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 35) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{3} - 2q^{4} - 2q^{9} + O(q^{10})$$ $$q + q^{3} - 2q^{4} - 2q^{9} - 3q^{11} - 2q^{12} + 5q^{13} + 4q^{16} + 3q^{17} - 2q^{19} + 6q^{23} - 5q^{27} + 3q^{29} + 4q^{31} - 3q^{33} + 4q^{36} - 2q^{37} + 5q^{39} + 12q^{41} + 10q^{43} + 6q^{44} + 9q^{47} + 4q^{48} + 3q^{51} - 10q^{52} - 12q^{53} - 2q^{57} - 8q^{61} - 8q^{64} + 4q^{67} - 6q^{68} + 6q^{69} + 2q^{73} + 4q^{76} - q^{79} + q^{81} + 12q^{83} + 3q^{87} + 12q^{89} - 12q^{92} + 4q^{93} - q^{97} + 6q^{99} + O(q^{100})$$

## 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
0 1.00000 −2.00000 0 0 0 0 −2.00000 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

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 1225.2.a.e 1
5.b even 2 1 245.2.a.c 1
5.c odd 4 2 1225.2.b.d 2
7.b odd 2 1 175.2.a.b 1
15.d odd 2 1 2205.2.a.e 1
20.d odd 2 1 3920.2.a.ba 1
21.c even 2 1 1575.2.a.f 1
28.d even 2 1 2800.2.a.z 1
35.c odd 2 1 35.2.a.a 1
35.f even 4 2 175.2.b.a 2
35.i odd 6 2 245.2.e.a 2
35.j even 6 2 245.2.e.b 2
105.g even 2 1 315.2.a.b 1
105.k odd 4 2 1575.2.d.c 2
140.c even 2 1 560.2.a.b 1
140.j odd 4 2 2800.2.g.l 2
280.c odd 2 1 2240.2.a.k 1
280.n even 2 1 2240.2.a.u 1
385.h even 2 1 4235.2.a.c 1
420.o odd 2 1 5040.2.a.v 1
455.h odd 2 1 5915.2.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.a.a 1 35.c odd 2 1
175.2.a.b 1 7.b odd 2 1
175.2.b.a 2 35.f even 4 2
245.2.a.c 1 5.b even 2 1
245.2.e.a 2 35.i odd 6 2
245.2.e.b 2 35.j even 6 2
315.2.a.b 1 105.g even 2 1
560.2.a.b 1 140.c even 2 1
1225.2.a.e 1 1.a even 1 1 trivial
1225.2.b.d 2 5.c odd 4 2
1575.2.a.f 1 21.c even 2 1
1575.2.d.c 2 105.k odd 4 2
2205.2.a.e 1 15.d odd 2 1
2240.2.a.k 1 280.c odd 2 1
2240.2.a.u 1 280.n even 2 1
2800.2.a.z 1 28.d even 2 1
2800.2.g.l 2 140.j odd 4 2
3920.2.a.ba 1 20.d odd 2 1
4235.2.a.c 1 385.h even 2 1
5040.2.a.v 1 420.o odd 2 1
5915.2.a.f 1 455.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1225))$$:

 $$T_{2}$$ $$T_{3} - 1$$

## Hecke characteristic polynomials

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