# Properties

 Label 3025.2.a.ba Level $3025$ Weight $2$ Character orbit 3025.a Self dual yes Analytic conductor $24.155$ Analytic rank $1$ Dimension $4$ CM no Inner twists $2$

# 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: $$1$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{3}, \sqrt{11})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 7x^{2} + 4$$ x^4 - 7*x^2 + 4 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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)

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_{2} - \beta_1) q^{3} + (\beta_{3} + 2) q^{4} - 2 q^{6} - 2 \beta_{2} q^{7} + (2 \beta_{2} + \beta_1) q^{8} - \beta_{3} q^{9}+O(q^{10})$$ q + b1 * q^2 + (b2 - b1) * q^3 + (b3 + 2) * q^4 - 2 * q^6 - 2*b2 * q^7 + (2*b2 + b1) * q^8 - b3 * q^9 $$q + \beta_1 q^{2} + (\beta_{2} - \beta_1) q^{3} + (\beta_{3} + 2) q^{4} - 2 q^{6} - 2 \beta_{2} q^{7} + (2 \beta_{2} + \beta_1) q^{8} - \beta_{3} q^{9} - 2 \beta_{2} q^{12} + ( - 2 \beta_{3} - 4) q^{14} + (\beta_{3} + 4) q^{16} + (2 \beta_{2} - 2 \beta_1) q^{17} + ( - 2 \beta_{2} - \beta_1) q^{18} - 4 q^{19} + (2 \beta_{3} - 2) q^{21} + ( - \beta_{2} + \beta_1) q^{23} - 2 \beta_{3} q^{24} + (\beta_{2} + \beta_1) q^{27} - 6 \beta_1 q^{28} + ( - 2 \beta_{3} - 4) q^{29} + (\beta_{3} + 1) q^{31} + ( - 2 \beta_{2} + 3 \beta_1) q^{32} - 4 q^{34} + ( - \beta_{3} - 8) q^{36} + ( - 5 \beta_{2} + 3 \beta_1) q^{37} - 4 \beta_1 q^{38} + ( - 2 \beta_{3} - 4) q^{41} + 4 \beta_{2} q^{42} - 2 \beta_{2} q^{43} + 2 q^{46} + (2 \beta_{2} - 4 \beta_1) q^{47} - 2 \beta_1 q^{48} + 5 q^{49} + ( - 2 \beta_{3} + 6) q^{51} + 4 \beta_1 q^{53} + (2 \beta_{3} + 6) q^{54} + ( - 2 \beta_{3} - 16) q^{56} + ( - 4 \beta_{2} + 4 \beta_1) q^{57} + ( - 4 \beta_{2} - 6 \beta_1) q^{58} + ( - \beta_{3} - 5) q^{59} + (2 \beta_{3} - 4) q^{61} + (2 \beta_{2} + 2 \beta_1) q^{62} + ( - 4 \beta_{2} + 6 \beta_1) q^{63} - \beta_{3} q^{64} + (\beta_{2} + 3 \beta_1) q^{67} - 4 \beta_{2} q^{68} + (\beta_{3} - 3) q^{69} + ( - 3 \beta_{3} - 3) q^{71} + (2 \beta_{2} - 7 \beta_1) q^{72} + 4 \beta_{2} q^{73} + ( - 2 \beta_{3} + 2) q^{74} + ( - 4 \beta_{3} - 8) q^{76} + (2 \beta_{3} - 6) q^{79} + (2 \beta_{3} - 1) q^{81} + ( - 4 \beta_{2} - 6 \beta_1) q^{82} + (2 \beta_{2} - 4 \beta_1) q^{83} + 12 q^{84} + ( - 2 \beta_{3} - 4) q^{86} + 4 \beta_{2} q^{87} + (\beta_{3} - 1) q^{89} + 2 \beta_{2} q^{92} + ( - 3 \beta_{2} + \beta_1) q^{93} + ( - 2 \beta_{3} - 12) q^{94} + (2 \beta_{3} - 8) q^{96} + ( - \beta_{2} + 3 \beta_1) q^{97} + 5 \beta_1 q^{98}+O(q^{100})$$ q + b1 * q^2 + (b2 - b1) * q^3 + (b3 + 2) * q^4 - 2 * q^6 - 2*b2 * q^7 + (2*b2 + b1) * q^8 - b3 * q^9 - 2*b2 * q^12 + (-2*b3 - 4) * q^14 + (b3 + 4) * q^16 + (2*b2 - 2*b1) * q^17 + (-2*b2 - b1) * q^18 - 4 * q^19 + (2*b3 - 2) * q^21 + (-b2 + b1) * q^23 - 2*b3 * q^24 + (b2 + b1) * q^27 - 6*b1 * q^28 + (-2*b3 - 4) * q^29 + (b3 + 1) * q^31 + (-2*b2 + 3*b1) * q^32 - 4 * q^34 + (-b3 - 8) * q^36 + (-5*b2 + 3*b1) * q^37 - 4*b1 * q^38 + (-2*b3 - 4) * q^41 + 4*b2 * q^42 - 2*b2 * q^43 + 2 * q^46 + (2*b2 - 4*b1) * q^47 - 2*b1 * q^48 + 5 * q^49 + (-2*b3 + 6) * q^51 + 4*b1 * q^53 + (2*b3 + 6) * q^54 + (-2*b3 - 16) * q^56 + (-4*b2 + 4*b1) * q^57 + (-4*b2 - 6*b1) * q^58 + (-b3 - 5) * q^59 + (2*b3 - 4) * q^61 + (2*b2 + 2*b1) * q^62 + (-4*b2 + 6*b1) * q^63 - b3 * q^64 + (b2 + 3*b1) * q^67 - 4*b2 * q^68 + (b3 - 3) * q^69 + (-3*b3 - 3) * q^71 + (2*b2 - 7*b1) * q^72 + 4*b2 * q^73 + (-2*b3 + 2) * q^74 + (-4*b3 - 8) * q^76 + (2*b3 - 6) * q^79 + (2*b3 - 1) * q^81 + (-4*b2 - 6*b1) * q^82 + (2*b2 - 4*b1) * q^83 + 12 * q^84 + (-2*b3 - 4) * q^86 + 4*b2 * q^87 + (b3 - 1) * q^89 + 2*b2 * q^92 + (-3*b2 + b1) * q^93 + (-2*b3 - 12) * q^94 + (2*b3 - 8) * q^96 + (-b2 + 3*b1) * q^97 + 5*b1 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 6 q^{4} - 8 q^{6} + 2 q^{9}+O(q^{10})$$ 4 * q + 6 * q^4 - 8 * q^6 + 2 * q^9 $$4 q + 6 q^{4} - 8 q^{6} + 2 q^{9} - 12 q^{14} + 14 q^{16} - 16 q^{19} - 12 q^{21} + 4 q^{24} - 12 q^{29} + 2 q^{31} - 16 q^{34} - 30 q^{36} - 12 q^{41} + 8 q^{46} + 20 q^{49} + 28 q^{51} + 20 q^{54} - 60 q^{56} - 18 q^{59} - 20 q^{61} + 2 q^{64} - 14 q^{69} - 6 q^{71} + 12 q^{74} - 24 q^{76} - 28 q^{79} - 8 q^{81} + 48 q^{84} - 12 q^{86} - 6 q^{89} - 44 q^{94} - 36 q^{96}+O(q^{100})$$ 4 * q + 6 * q^4 - 8 * q^6 + 2 * q^9 - 12 * q^14 + 14 * q^16 - 16 * q^19 - 12 * q^21 + 4 * q^24 - 12 * q^29 + 2 * q^31 - 16 * q^34 - 30 * q^36 - 12 * q^41 + 8 * q^46 + 20 * q^49 + 28 * q^51 + 20 * q^54 - 60 * q^56 - 18 * q^59 - 20 * q^61 + 2 * q^64 - 14 * q^69 - 6 * q^71 + 12 * q^74 - 24 * q^76 - 28 * q^79 - 8 * q^81 + 48 * q^84 - 12 * q^86 - 6 * q^89 - 44 * q^94 - 36 * q^96

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 7x^{2} + 4$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} - 5\nu ) / 2$$ (v^3 - 5*v) / 2 $$\beta_{3}$$ $$=$$ $$\nu^{2} - 4$$ v^2 - 4
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + 4$$ b3 + 4 $$\nu^{3}$$ $$=$$ $$2\beta_{2} + 5\beta_1$$ 2*b2 + 5*b1

## 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
 −2.52434 −0.792287 0.792287 2.52434
−2.52434 0.792287 4.37228 0 −2.00000 3.46410 −5.98844 −2.37228 0
1.2 −0.792287 2.52434 −1.37228 0 −2.00000 −3.46410 2.67181 3.37228 0
1.3 0.792287 −2.52434 −1.37228 0 −2.00000 3.46410 −2.67181 3.37228 0
1.4 2.52434 −0.792287 4.37228 0 −2.00000 −3.46410 5.98844 −2.37228 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

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3025.2.a.ba 4
5.b even 2 1 inner 3025.2.a.ba 4
5.c odd 4 2 605.2.b.c 4
11.b odd 2 1 275.2.a.h 4
33.d even 2 1 2475.2.a.bi 4
44.c even 2 1 4400.2.a.cc 4
55.d odd 2 1 275.2.a.h 4
55.e even 4 2 55.2.b.a 4
55.k odd 20 8 605.2.j.j 16
55.l even 20 8 605.2.j.i 16
165.d even 2 1 2475.2.a.bi 4
165.l odd 4 2 495.2.c.a 4
220.g even 2 1 4400.2.a.cc 4
220.i odd 4 2 880.2.b.h 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.b.a 4 55.e even 4 2
275.2.a.h 4 11.b odd 2 1
275.2.a.h 4 55.d odd 2 1
495.2.c.a 4 165.l odd 4 2
605.2.b.c 4 5.c odd 4 2
605.2.j.i 16 55.l even 20 8
605.2.j.j 16 55.k odd 20 8
880.2.b.h 4 220.i odd 4 2
2475.2.a.bi 4 33.d even 2 1
2475.2.a.bi 4 165.d even 2 1
3025.2.a.ba 4 1.a even 1 1 trivial
3025.2.a.ba 4 5.b even 2 1 inner
4400.2.a.cc 4 44.c even 2 1
4400.2.a.cc 4 220.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3025))$$:

 $$T_{2}^{4} - 7T_{2}^{2} + 4$$ T2^4 - 7*T2^2 + 4 $$T_{3}^{4} - 7T_{3}^{2} + 4$$ T3^4 - 7*T3^2 + 4 $$T_{19} + 4$$ T19 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 7T^{2} + 4$$
$3$ $$T^{4} - 7T^{2} + 4$$
$5$ $$T^{4}$$
$7$ $$(T^{2} - 12)^{2}$$
$11$ $$T^{4}$$
$13$ $$T^{4}$$
$17$ $$T^{4} - 28T^{2} + 64$$
$19$ $$(T + 4)^{4}$$
$23$ $$T^{4} - 7T^{2} + 4$$
$29$ $$(T^{2} + 6 T - 24)^{2}$$
$31$ $$(T^{2} - T - 8)^{2}$$
$37$ $$T^{4} - 123T^{2} + 144$$
$41$ $$(T^{2} + 6 T - 24)^{2}$$
$43$ $$(T^{2} - 12)^{2}$$
$47$ $$(T^{2} - 44)^{2}$$
$53$ $$T^{4} - 112T^{2} + 1024$$
$59$ $$(T^{2} + 9 T + 12)^{2}$$
$61$ $$(T^{2} + 10 T - 8)^{2}$$
$67$ $$T^{4} - 87T^{2} + 36$$
$71$ $$(T^{2} + 3 T - 72)^{2}$$
$73$ $$(T^{2} - 48)^{2}$$
$79$ $$(T^{2} + 14 T + 16)^{2}$$
$83$ $$(T^{2} - 44)^{2}$$
$89$ $$(T^{2} + 3 T - 6)^{2}$$
$97$ $$T^{4} - 51T^{2} + 576$$