# Properties

 Label 3025.2.a.n Level $3025$ Weight $2$ Character orbit 3025.a Self dual yes Analytic conductor $24.155$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{13})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 3$$ x^2 - x - 3 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 275) 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)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{13})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + (\beta - 1) q^{3} + (\beta + 1) q^{4} + 3 q^{6} + (\beta + 2) q^{7} + 3 q^{8} + ( - \beta + 1) q^{9}+O(q^{10})$$ q + b * q^2 + (b - 1) * q^3 + (b + 1) * q^4 + 3 * q^6 + (b + 2) * q^7 + 3 * q^8 + (-b + 1) * q^9 $$q + \beta q^{2} + (\beta - 1) q^{3} + (\beta + 1) q^{4} + 3 q^{6} + (\beta + 2) q^{7} + 3 q^{8} + ( - \beta + 1) q^{9} + (\beta + 2) q^{12} + 5 q^{13} + (3 \beta + 3) q^{14} + (\beta - 2) q^{16} + ( - 3 \beta + 3) q^{17} - 3 q^{18} + q^{19} + (2 \beta + 1) q^{21} + ( - \beta + 6) q^{23} + (3 \beta - 3) q^{24} + 5 \beta q^{26} + ( - 2 \beta - 1) q^{27} + (4 \beta + 5) q^{28} + (3 \beta + 3) q^{29} + ( - 4 \beta + 5) q^{31} + ( - \beta - 3) q^{32} - 9 q^{34} + ( - \beta - 2) q^{36} + ( - 2 \beta - 5) q^{37} + \beta q^{38} + (5 \beta - 5) q^{39} + ( - 2 \beta + 3) q^{41} + (3 \beta + 6) q^{42} + ( - 4 \beta + 2) q^{43} + (5 \beta - 3) q^{46} + 3 q^{47} + ( - 2 \beta + 5) q^{48} + 5 \beta q^{49} + (3 \beta - 12) q^{51} + (5 \beta + 5) q^{52} - \beta q^{53} + ( - 3 \beta - 6) q^{54} + (3 \beta + 6) q^{56} + (\beta - 1) q^{57} + (6 \beta + 9) q^{58} + (4 \beta - 9) q^{59} + ( - 3 \beta + 4) q^{61} + (\beta - 12) q^{62} + ( - 2 \beta - 1) q^{63} + ( - 6 \beta + 1) q^{64} + 4 q^{67} + ( - 3 \beta - 6) q^{68} + (6 \beta - 9) q^{69} + 2 \beta q^{71} + ( - 3 \beta + 3) q^{72} + (3 \beta - 4) q^{73} + ( - 7 \beta - 6) q^{74} + (\beta + 1) q^{76} + 15 q^{78} + ( - 3 \beta + 7) q^{79} + (2 \beta - 8) q^{81} + (\beta - 6) q^{82} + (5 \beta + 3) q^{83} + (5 \beta + 7) q^{84} + ( - 2 \beta - 12) q^{86} + (3 \beta + 6) q^{87} + (\beta + 3) q^{89} + (5 \beta + 10) q^{91} + (4 \beta + 3) q^{92} + (5 \beta - 17) q^{93} + 3 \beta q^{94} - 3 \beta q^{96} + (\beta - 14) q^{97} + (5 \beta + 15) q^{98} +O(q^{100})$$ q + b * q^2 + (b - 1) * q^3 + (b + 1) * q^4 + 3 * q^6 + (b + 2) * q^7 + 3 * q^8 + (-b + 1) * q^9 + (b + 2) * q^12 + 5 * q^13 + (3*b + 3) * q^14 + (b - 2) * q^16 + (-3*b + 3) * q^17 - 3 * q^18 + q^19 + (2*b + 1) * q^21 + (-b + 6) * q^23 + (3*b - 3) * q^24 + 5*b * q^26 + (-2*b - 1) * q^27 + (4*b + 5) * q^28 + (3*b + 3) * q^29 + (-4*b + 5) * q^31 + (-b - 3) * q^32 - 9 * q^34 + (-b - 2) * q^36 + (-2*b - 5) * q^37 + b * q^38 + (5*b - 5) * q^39 + (-2*b + 3) * q^41 + (3*b + 6) * q^42 + (-4*b + 2) * q^43 + (5*b - 3) * q^46 + 3 * q^47 + (-2*b + 5) * q^48 + 5*b * q^49 + (3*b - 12) * q^51 + (5*b + 5) * q^52 - b * q^53 + (-3*b - 6) * q^54 + (3*b + 6) * q^56 + (b - 1) * q^57 + (6*b + 9) * q^58 + (4*b - 9) * q^59 + (-3*b + 4) * q^61 + (b - 12) * q^62 + (-2*b - 1) * q^63 + (-6*b + 1) * q^64 + 4 * q^67 + (-3*b - 6) * q^68 + (6*b - 9) * q^69 + 2*b * q^71 + (-3*b + 3) * q^72 + (3*b - 4) * q^73 + (-7*b - 6) * q^74 + (b + 1) * q^76 + 15 * q^78 + (-3*b + 7) * q^79 + (2*b - 8) * q^81 + (b - 6) * q^82 + (5*b + 3) * q^83 + (5*b + 7) * q^84 + (-2*b - 12) * q^86 + (3*b + 6) * q^87 + (b + 3) * q^89 + (5*b + 10) * q^91 + (4*b + 3) * q^92 + (5*b - 17) * q^93 + 3*b * q^94 - 3*b * q^96 + (b - 14) * q^97 + (5*b + 15) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} - q^{3} + 3 q^{4} + 6 q^{6} + 5 q^{7} + 6 q^{8} + q^{9}+O(q^{10})$$ 2 * q + q^2 - q^3 + 3 * q^4 + 6 * q^6 + 5 * q^7 + 6 * q^8 + q^9 $$2 q + q^{2} - q^{3} + 3 q^{4} + 6 q^{6} + 5 q^{7} + 6 q^{8} + q^{9} + 5 q^{12} + 10 q^{13} + 9 q^{14} - 3 q^{16} + 3 q^{17} - 6 q^{18} + 2 q^{19} + 4 q^{21} + 11 q^{23} - 3 q^{24} + 5 q^{26} - 4 q^{27} + 14 q^{28} + 9 q^{29} + 6 q^{31} - 7 q^{32} - 18 q^{34} - 5 q^{36} - 12 q^{37} + q^{38} - 5 q^{39} + 4 q^{41} + 15 q^{42} - q^{46} + 6 q^{47} + 8 q^{48} + 5 q^{49} - 21 q^{51} + 15 q^{52} - q^{53} - 15 q^{54} + 15 q^{56} - q^{57} + 24 q^{58} - 14 q^{59} + 5 q^{61} - 23 q^{62} - 4 q^{63} - 4 q^{64} + 8 q^{67} - 15 q^{68} - 12 q^{69} + 2 q^{71} + 3 q^{72} - 5 q^{73} - 19 q^{74} + 3 q^{76} + 30 q^{78} + 11 q^{79} - 14 q^{81} - 11 q^{82} + 11 q^{83} + 19 q^{84} - 26 q^{86} + 15 q^{87} + 7 q^{89} + 25 q^{91} + 10 q^{92} - 29 q^{93} + 3 q^{94} - 3 q^{96} - 27 q^{97} + 35 q^{98}+O(q^{100})$$ 2 * q + q^2 - q^3 + 3 * q^4 + 6 * q^6 + 5 * q^7 + 6 * q^8 + q^9 + 5 * q^12 + 10 * q^13 + 9 * q^14 - 3 * q^16 + 3 * q^17 - 6 * q^18 + 2 * q^19 + 4 * q^21 + 11 * q^23 - 3 * q^24 + 5 * q^26 - 4 * q^27 + 14 * q^28 + 9 * q^29 + 6 * q^31 - 7 * q^32 - 18 * q^34 - 5 * q^36 - 12 * q^37 + q^38 - 5 * q^39 + 4 * q^41 + 15 * q^42 - q^46 + 6 * q^47 + 8 * q^48 + 5 * q^49 - 21 * q^51 + 15 * q^52 - q^53 - 15 * q^54 + 15 * q^56 - q^57 + 24 * q^58 - 14 * q^59 + 5 * q^61 - 23 * q^62 - 4 * q^63 - 4 * q^64 + 8 * q^67 - 15 * q^68 - 12 * q^69 + 2 * q^71 + 3 * q^72 - 5 * q^73 - 19 * q^74 + 3 * q^76 + 30 * q^78 + 11 * q^79 - 14 * q^81 - 11 * q^82 + 11 * q^83 + 19 * q^84 - 26 * q^86 + 15 * q^87 + 7 * q^89 + 25 * q^91 + 10 * q^92 - 29 * q^93 + 3 * q^94 - 3 * q^96 - 27 * q^97 + 35 * 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
 −1.30278 2.30278
−1.30278 −2.30278 −0.302776 0 3.00000 0.697224 3.00000 2.30278 0
1.2 2.30278 1.30278 3.30278 0 3.00000 4.30278 3.00000 −1.30278 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.n 2
5.b even 2 1 3025.2.a.h 2
11.b odd 2 1 275.2.a.e 2
33.d even 2 1 2475.2.a.t 2
44.c even 2 1 4400.2.a.bs 2
55.d odd 2 1 275.2.a.f yes 2
55.e even 4 2 275.2.b.c 4
165.d even 2 1 2475.2.a.o 2
165.l odd 4 2 2475.2.c.k 4
220.g even 2 1 4400.2.a.bh 2
220.i odd 4 2 4400.2.b.y 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
275.2.a.e 2 11.b odd 2 1
275.2.a.f yes 2 55.d odd 2 1
275.2.b.c 4 55.e even 4 2
2475.2.a.o 2 165.d even 2 1
2475.2.a.t 2 33.d even 2 1
2475.2.c.k 4 165.l odd 4 2
3025.2.a.h 2 5.b even 2 1
3025.2.a.n 2 1.a even 1 1 trivial
4400.2.a.bh 2 220.g even 2 1
4400.2.a.bs 2 44.c even 2 1
4400.2.b.y 4 220.i odd 4 2

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T - 3$$
$3$ $$T^{2} + T - 3$$
$5$ $$T^{2}$$
$7$ $$T^{2} - 5T + 3$$
$11$ $$T^{2}$$
$13$ $$(T - 5)^{2}$$
$17$ $$T^{2} - 3T - 27$$
$19$ $$(T - 1)^{2}$$
$23$ $$T^{2} - 11T + 27$$
$29$ $$T^{2} - 9T - 9$$
$31$ $$T^{2} - 6T - 43$$
$37$ $$T^{2} + 12T + 23$$
$41$ $$T^{2} - 4T - 9$$
$43$ $$T^{2} - 52$$
$47$ $$(T - 3)^{2}$$
$53$ $$T^{2} + T - 3$$
$59$ $$T^{2} + 14T - 3$$
$61$ $$T^{2} - 5T - 23$$
$67$ $$(T - 4)^{2}$$
$71$ $$T^{2} - 2T - 12$$
$73$ $$T^{2} + 5T - 23$$
$79$ $$T^{2} - 11T + 1$$
$83$ $$T^{2} - 11T - 51$$
$89$ $$T^{2} - 7T + 9$$
$97$ $$T^{2} + 27T + 179$$