# Properties

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

# Related objects

## 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: $$1$$ 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

 $$f(q)$$ $$=$$ $$q - 2 q^{2} + q^{3} + 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10})$$ q - 2 * q^2 + q^3 + 2 * q^4 - 2 * q^6 - 2 * q^9 $$q - 2 q^{2} + q^{3} + 2 q^{4} - 2 q^{6} - 2 q^{9} - 3 q^{11} + 2 q^{12} + q^{13} - 4 q^{16} + 7 q^{17} + 4 q^{18} + 6 q^{22} - 6 q^{23} - 2 q^{26} - 5 q^{27} - 5 q^{29} - 2 q^{31} + 8 q^{32} - 3 q^{33} - 14 q^{34} - 4 q^{36} - 2 q^{37} + q^{39} - 2 q^{41} + 4 q^{43} - 6 q^{44} + 12 q^{46} - 3 q^{47} - 4 q^{48} + 7 q^{51} + 2 q^{52} - 6 q^{53} + 10 q^{54} + 10 q^{58} - 10 q^{59} + 8 q^{61} + 4 q^{62} - 8 q^{64} + 6 q^{66} - 2 q^{67} + 14 q^{68} - 6 q^{69} - 8 q^{71} + 6 q^{73} + 4 q^{74} - 2 q^{78} - 5 q^{79} + q^{81} + 4 q^{82} - 4 q^{83} - 8 q^{86} - 5 q^{87} - 12 q^{92} - 2 q^{93} + 6 q^{94} + 8 q^{96} + 7 q^{97} + 6 q^{99}+O(q^{100})$$ q - 2 * q^2 + q^3 + 2 * q^4 - 2 * q^6 - 2 * q^9 - 3 * q^11 + 2 * q^12 + q^13 - 4 * q^16 + 7 * q^17 + 4 * q^18 + 6 * q^22 - 6 * q^23 - 2 * q^26 - 5 * q^27 - 5 * q^29 - 2 * q^31 + 8 * q^32 - 3 * q^33 - 14 * q^34 - 4 * q^36 - 2 * q^37 + q^39 - 2 * q^41 + 4 * q^43 - 6 * q^44 + 12 * q^46 - 3 * q^47 - 4 * q^48 + 7 * q^51 + 2 * q^52 - 6 * q^53 + 10 * q^54 + 10 * q^58 - 10 * q^59 + 8 * q^61 + 4 * q^62 - 8 * q^64 + 6 * q^66 - 2 * q^67 + 14 * q^68 - 6 * q^69 - 8 * q^71 + 6 * q^73 + 4 * q^74 - 2 * q^78 - 5 * q^79 + q^81 + 4 * q^82 - 4 * q^83 - 8 * q^86 - 5 * q^87 - 12 * q^92 - 2 * q^93 + 6 * q^94 + 8 * q^96 + 7 * q^97 + 6 * q^99

## 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
−2.00000 1.00000 2.00000 0 −2.00000 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.a 1
5.b even 2 1 1225.2.a.i 1
5.c odd 4 2 245.2.b.a 2
7.b odd 2 1 175.2.a.a 1
15.e even 4 2 2205.2.d.b 2
21.c even 2 1 1575.2.a.k 1
28.d even 2 1 2800.2.a.w 1
35.c odd 2 1 175.2.a.c 1
35.f even 4 2 35.2.b.a 2
35.k even 12 4 245.2.j.e 4
35.l odd 12 4 245.2.j.d 4
105.g even 2 1 1575.2.a.a 1
105.k odd 4 2 315.2.d.a 2
140.c even 2 1 2800.2.a.l 1
140.j odd 4 2 560.2.g.b 2
280.s even 4 2 2240.2.g.h 2
280.y odd 4 2 2240.2.g.g 2
420.w even 4 2 5040.2.t.p 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.b.a 2 35.f even 4 2
175.2.a.a 1 7.b odd 2 1
175.2.a.c 1 35.c odd 2 1
245.2.b.a 2 5.c odd 4 2
245.2.j.d 4 35.l odd 12 4
245.2.j.e 4 35.k even 12 4
315.2.d.a 2 105.k odd 4 2
560.2.g.b 2 140.j odd 4 2
1225.2.a.a 1 1.a even 1 1 trivial
1225.2.a.i 1 5.b even 2 1
1575.2.a.a 1 105.g even 2 1
1575.2.a.k 1 21.c even 2 1
2205.2.d.b 2 15.e even 4 2
2240.2.g.g 2 280.y odd 4 2
2240.2.g.h 2 280.s even 4 2
2800.2.a.l 1 140.c even 2 1
2800.2.a.w 1 28.d even 2 1
5040.2.t.p 2 420.w even 4 2

## Hecke kernels

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

 $$T_{2} + 2$$ T2 + 2 $$T_{3} - 1$$ T3 - 1

## Hecke characteristic polynomials

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