# Properties

 Label 3025.2.a.b 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 121) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{2} - 2q^{3} - q^{4} + 2q^{6} + 2q^{7} + 3q^{8} + q^{9} + O(q^{10})$$ $$q - q^{2} - 2q^{3} - q^{4} + 2q^{6} + 2q^{7} + 3q^{8} + q^{9} + 2q^{12} - q^{13} - 2q^{14} - q^{16} + 5q^{17} - q^{18} + 6q^{19} - 4q^{21} - 2q^{23} - 6q^{24} + q^{26} + 4q^{27} - 2q^{28} + 9q^{29} - 2q^{31} - 5q^{32} - 5q^{34} - q^{36} + 3q^{37} - 6q^{38} + 2q^{39} - 5q^{41} + 4q^{42} + 2q^{46} - 2q^{47} + 2q^{48} - 3q^{49} - 10q^{51} + q^{52} - 9q^{53} - 4q^{54} + 6q^{56} - 12q^{57} - 9q^{58} + 8q^{59} + 6q^{61} + 2q^{62} + 2q^{63} + 7q^{64} - 2q^{67} - 5q^{68} + 4q^{69} + 12q^{71} + 3q^{72} + 2q^{73} - 3q^{74} - 6q^{76} - 2q^{78} - 10q^{79} - 11q^{81} + 5q^{82} - 6q^{83} + 4q^{84} - 18q^{87} - 9q^{89} - 2q^{91} + 2q^{92} + 4q^{93} + 2q^{94} + 10q^{96} + 13q^{97} + 3q^{98} + O(q^{100})$$

## 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 −2.00000 −1.00000 0 2.00000 2.00000 3.00000 1.00000 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.b 1
5.b even 2 1 121.2.a.c yes 1
11.b odd 2 1 3025.2.a.e 1
15.d odd 2 1 1089.2.a.c 1
20.d odd 2 1 1936.2.a.b 1
35.c odd 2 1 5929.2.a.g 1
40.e odd 2 1 7744.2.a.bf 1
40.f even 2 1 7744.2.a.c 1
55.d odd 2 1 121.2.a.a 1
55.h odd 10 4 121.2.c.d 4
55.j even 10 4 121.2.c.b 4
165.d even 2 1 1089.2.a.i 1
220.g even 2 1 1936.2.a.a 1
385.h even 2 1 5929.2.a.a 1
440.c even 2 1 7744.2.a.be 1
440.o odd 2 1 7744.2.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
121.2.a.a 1 55.d odd 2 1
121.2.a.c yes 1 5.b even 2 1
121.2.c.b 4 55.j even 10 4
121.2.c.d 4 55.h odd 10 4
1089.2.a.c 1 15.d odd 2 1
1089.2.a.i 1 165.d even 2 1
1936.2.a.a 1 220.g even 2 1
1936.2.a.b 1 20.d odd 2 1
3025.2.a.b 1 1.a even 1 1 trivial
3025.2.a.e 1 11.b odd 2 1
5929.2.a.a 1 385.h even 2 1
5929.2.a.g 1 35.c odd 2 1
7744.2.a.c 1 40.f even 2 1
7744.2.a.f 1 440.o odd 2 1
7744.2.a.be 1 440.c even 2 1
7744.2.a.bf 1 40.e odd 2 1

## Hecke kernels

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

 $$T_{2} + 1$$ $$T_{3} + 2$$ $$T_{19} - 6$$

## Hecke characteristic polynomials

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