# Properties

 Label 3025.2.a.c Level $3025$ Weight $2$ Character orbit 3025.a Self dual yes Analytic conductor $24.155$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3025 = 5^{2} \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3025.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$24.1547466114$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 605) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{2} + 3q^{3} - q^{4} - 3q^{6} - 3q^{7} + 3q^{8} + 6q^{9} + O(q^{10})$$ $$q - q^{2} + 3q^{3} - q^{4} - 3q^{6} - 3q^{7} + 3q^{8} + 6q^{9} - 3q^{12} + 4q^{13} + 3q^{14} - q^{16} - 6q^{18} - 4q^{19} - 9q^{21} + 8q^{23} + 9q^{24} - 4q^{26} + 9q^{27} + 3q^{28} - 6q^{29} - 2q^{31} - 5q^{32} - 6q^{36} + 8q^{37} + 4q^{38} + 12q^{39} + 5q^{41} + 9q^{42} + 5q^{43} - 8q^{46} + 3q^{47} - 3q^{48} + 2q^{49} - 4q^{52} - 4q^{53} - 9q^{54} - 9q^{56} - 12q^{57} + 6q^{58} - 2q^{59} + 11q^{61} + 2q^{62} - 18q^{63} + 7q^{64} + 13q^{67} + 24q^{69} + 2q^{71} + 18q^{72} - 8q^{73} - 8q^{74} + 4q^{76} - 12q^{78} - 10q^{79} + 9q^{81} - 5q^{82} + 4q^{83} + 9q^{84} - 5q^{86} - 18q^{87} + q^{89} - 12q^{91} - 8q^{92} - 6q^{93} - 3q^{94} - 15q^{96} + 8q^{97} - 2q^{98} + O(q^{100})$$

## 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
 0
−1.00000 3.00000 −1.00000 0 −3.00000 −3.00000 3.00000 6.00000 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
$$5$$ $$1$$
$$11$$ $$-1$$

## 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 3025.2.a.c 1
5.b even 2 1 605.2.a.c yes 1
11.b odd 2 1 3025.2.a.g 1
15.d odd 2 1 5445.2.a.d 1
20.d odd 2 1 9680.2.a.be 1
55.d odd 2 1 605.2.a.a 1
55.h odd 10 4 605.2.g.d 4
55.j even 10 4 605.2.g.b 4
165.d even 2 1 5445.2.a.h 1
220.g even 2 1 9680.2.a.bf 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
605.2.a.a 1 55.d odd 2 1
605.2.a.c yes 1 5.b even 2 1
605.2.g.b 4 55.j even 10 4
605.2.g.d 4 55.h odd 10 4
3025.2.a.c 1 1.a even 1 1 trivial
3025.2.a.g 1 11.b odd 2 1
5445.2.a.d 1 15.d odd 2 1
5445.2.a.h 1 165.d even 2 1
9680.2.a.be 1 20.d odd 2 1
9680.2.a.bf 1 220.g even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3025))$$:

 $$T_{2} + 1$$ $$T_{3} - 3$$ $$T_{19} + 4$$

## Hecke characteristic polynomials

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