# Properties

 Label 3025.2.a.f 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

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3025,2,Mod(1,3025)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3025, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 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 55) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{4} - 3 q^{8} - 3 q^{9}+O(q^{10})$$ q + q^2 - q^4 - 3 * q^8 - 3 * q^9 $$q + q^{2} - q^{4} - 3 q^{8} - 3 q^{9} + 2 q^{13} - q^{16} + 6 q^{17} - 3 q^{18} + 4 q^{19} - 4 q^{23} + 2 q^{26} - 6 q^{29} - 8 q^{31} + 5 q^{32} + 6 q^{34} + 3 q^{36} + 2 q^{37} + 4 q^{38} - 2 q^{41} + 4 q^{43} - 4 q^{46} + 12 q^{47} - 7 q^{49} - 2 q^{52} + 2 q^{53} - 6 q^{58} + 4 q^{59} + 10 q^{61} - 8 q^{62} + 7 q^{64} + 16 q^{67} - 6 q^{68} + 8 q^{71} + 9 q^{72} + 14 q^{73} + 2 q^{74} - 4 q^{76} - 8 q^{79} + 9 q^{81} - 2 q^{82} - 4 q^{83} + 4 q^{86} + 10 q^{89} + 4 q^{92} + 12 q^{94} - 10 q^{97} - 7 q^{98}+O(q^{100})$$ q + q^2 - q^4 - 3 * q^8 - 3 * q^9 + 2 * q^13 - q^16 + 6 * q^17 - 3 * q^18 + 4 * q^19 - 4 * q^23 + 2 * q^26 - 6 * q^29 - 8 * q^31 + 5 * q^32 + 6 * q^34 + 3 * q^36 + 2 * q^37 + 4 * q^38 - 2 * q^41 + 4 * q^43 - 4 * q^46 + 12 * q^47 - 7 * q^49 - 2 * q^52 + 2 * q^53 - 6 * q^58 + 4 * q^59 + 10 * q^61 - 8 * q^62 + 7 * q^64 + 16 * q^67 - 6 * q^68 + 8 * q^71 + 9 * q^72 + 14 * q^73 + 2 * q^74 - 4 * q^76 - 8 * q^79 + 9 * q^81 - 2 * q^82 - 4 * q^83 + 4 * q^86 + 10 * q^89 + 4 * q^92 + 12 * q^94 - 10 * q^97 - 7 * q^98

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

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 0 −1.00000 0 0 0 −3.00000 −3.00000 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.f 1
5.b even 2 1 605.2.a.b 1
11.b odd 2 1 275.2.a.a 1
15.d odd 2 1 5445.2.a.i 1
20.d odd 2 1 9680.2.a.r 1
33.d even 2 1 2475.2.a.i 1
44.c even 2 1 4400.2.a.p 1
55.d odd 2 1 55.2.a.a 1
55.e even 4 2 275.2.b.b 2
55.h odd 10 4 605.2.g.a 4
55.j even 10 4 605.2.g.c 4
165.d even 2 1 495.2.a.a 1
165.l odd 4 2 2475.2.c.f 2
220.g even 2 1 880.2.a.h 1
220.i odd 4 2 4400.2.b.n 2
385.h even 2 1 2695.2.a.c 1
440.c even 2 1 3520.2.a.n 1
440.o odd 2 1 3520.2.a.p 1
660.g odd 2 1 7920.2.a.i 1
715.c odd 2 1 9295.2.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.a.a 1 55.d odd 2 1
275.2.a.a 1 11.b odd 2 1
275.2.b.b 2 55.e even 4 2
495.2.a.a 1 165.d even 2 1
605.2.a.b 1 5.b even 2 1
605.2.g.a 4 55.h odd 10 4
605.2.g.c 4 55.j even 10 4
880.2.a.h 1 220.g even 2 1
2475.2.a.i 1 33.d even 2 1
2475.2.c.f 2 165.l odd 4 2
2695.2.a.c 1 385.h even 2 1
3025.2.a.f 1 1.a even 1 1 trivial
3520.2.a.n 1 440.c even 2 1
3520.2.a.p 1 440.o odd 2 1
4400.2.a.p 1 44.c even 2 1
4400.2.b.n 2 220.i odd 4 2
5445.2.a.i 1 15.d odd 2 1
7920.2.a.i 1 660.g odd 2 1
9295.2.a.b 1 715.c odd 2 1
9680.2.a.r 1 20.d odd 2 1

## Hecke kernels

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

 $$T_{2} - 1$$ T2 - 1 $$T_{3}$$ T3 $$T_{19} - 4$$ T19 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T$$
$11$ $$T$$
$13$ $$T - 2$$
$17$ $$T - 6$$
$19$ $$T - 4$$
$23$ $$T + 4$$
$29$ $$T + 6$$
$31$ $$T + 8$$
$37$ $$T - 2$$
$41$ $$T + 2$$
$43$ $$T - 4$$
$47$ $$T - 12$$
$53$ $$T - 2$$
$59$ $$T - 4$$
$61$ $$T - 10$$
$67$ $$T - 16$$
$71$ $$T - 8$$
$73$ $$T - 14$$
$79$ $$T + 8$$
$83$ $$T + 4$$
$89$ $$T - 10$$
$97$ $$T + 10$$