# Properties

 Label 225.2.e.a Level $225$ Weight $2$ Character orbit 225.e Analytic conductor $1.797$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,2,Mod(76,225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(225, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([2, 0]))

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

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

chi := DirichletCharacter("225.76");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$225 = 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 225.e (of order $$3$$, degree $$2$$, minimal)

## 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: no Analytic conductor: $$1.79663404548$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 45) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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 primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{6} + 1) q^{2} + ( - \zeta_{6} + 2) q^{3} + \zeta_{6} q^{4} + ( - 2 \zeta_{6} + 1) q^{6} + (3 \zeta_{6} - 3) q^{7} + 3 q^{8} + ( - 3 \zeta_{6} + 3) q^{9}+O(q^{10})$$ q + (-z + 1) * q^2 + (-z + 2) * q^3 + z * q^4 + (-2*z + 1) * q^6 + (3*z - 3) * q^7 + 3 * q^8 + (-3*z + 3) * q^9 $$q + ( - \zeta_{6} + 1) q^{2} + ( - \zeta_{6} + 2) q^{3} + \zeta_{6} q^{4} + ( - 2 \zeta_{6} + 1) q^{6} + (3 \zeta_{6} - 3) q^{7} + 3 q^{8} + ( - 3 \zeta_{6} + 3) q^{9} + ( - 2 \zeta_{6} + 2) q^{11} + (\zeta_{6} + 1) q^{12} - 2 \zeta_{6} q^{13} + 3 \zeta_{6} q^{14} + ( - \zeta_{6} + 1) q^{16} - 4 q^{17} - 3 \zeta_{6} q^{18} - 8 q^{19} + (6 \zeta_{6} - 3) q^{21} - 2 \zeta_{6} q^{22} + 3 \zeta_{6} q^{23} + ( - 3 \zeta_{6} + 6) q^{24} - 2 q^{26} + ( - 6 \zeta_{6} + 3) q^{27} - 3 q^{28} + ( - \zeta_{6} + 1) q^{29} + 5 \zeta_{6} q^{32} + ( - 4 \zeta_{6} + 2) q^{33} + (4 \zeta_{6} - 4) q^{34} + 3 q^{36} + 4 q^{37} + (8 \zeta_{6} - 8) q^{38} + ( - 2 \zeta_{6} - 2) q^{39} - 5 \zeta_{6} q^{41} + (3 \zeta_{6} + 3) q^{42} + (8 \zeta_{6} - 8) q^{43} + 2 q^{44} + 3 q^{46} + ( - 7 \zeta_{6} + 7) q^{47} + ( - 2 \zeta_{6} + 1) q^{48} - 2 \zeta_{6} q^{49} + (4 \zeta_{6} - 8) q^{51} + ( - 2 \zeta_{6} + 2) q^{52} + 2 q^{53} + ( - 3 \zeta_{6} - 3) q^{54} + (9 \zeta_{6} - 9) q^{56} + (8 \zeta_{6} - 16) q^{57} - \zeta_{6} q^{58} + 14 \zeta_{6} q^{59} + (7 \zeta_{6} - 7) q^{61} + 9 \zeta_{6} q^{63} + 7 q^{64} + ( - 2 \zeta_{6} - 2) q^{66} - 3 \zeta_{6} q^{67} - 4 \zeta_{6} q^{68} + (3 \zeta_{6} + 3) q^{69} + 2 q^{71} + ( - 9 \zeta_{6} + 9) q^{72} - 4 q^{73} + ( - 4 \zeta_{6} + 4) q^{74} - 8 \zeta_{6} q^{76} + 6 \zeta_{6} q^{77} + (2 \zeta_{6} - 4) q^{78} + ( - 6 \zeta_{6} + 6) q^{79} - 9 \zeta_{6} q^{81} - 5 q^{82} + ( - 9 \zeta_{6} + 9) q^{83} + (3 \zeta_{6} - 6) q^{84} + 8 \zeta_{6} q^{86} + ( - 2 \zeta_{6} + 1) q^{87} + ( - 6 \zeta_{6} + 6) q^{88} - 15 q^{89} + 6 q^{91} + (3 \zeta_{6} - 3) q^{92} - 7 \zeta_{6} q^{94} + (5 \zeta_{6} + 5) q^{96} + ( - 2 \zeta_{6} + 2) q^{97} - 2 q^{98} - 6 \zeta_{6} q^{99} +O(q^{100})$$ q + (-z + 1) * q^2 + (-z + 2) * q^3 + z * q^4 + (-2*z + 1) * q^6 + (3*z - 3) * q^7 + 3 * q^8 + (-3*z + 3) * q^9 + (-2*z + 2) * q^11 + (z + 1) * q^12 - 2*z * q^13 + 3*z * q^14 + (-z + 1) * q^16 - 4 * q^17 - 3*z * q^18 - 8 * q^19 + (6*z - 3) * q^21 - 2*z * q^22 + 3*z * q^23 + (-3*z + 6) * q^24 - 2 * q^26 + (-6*z + 3) * q^27 - 3 * q^28 + (-z + 1) * q^29 + 5*z * q^32 + (-4*z + 2) * q^33 + (4*z - 4) * q^34 + 3 * q^36 + 4 * q^37 + (8*z - 8) * q^38 + (-2*z - 2) * q^39 - 5*z * q^41 + (3*z + 3) * q^42 + (8*z - 8) * q^43 + 2 * q^44 + 3 * q^46 + (-7*z + 7) * q^47 + (-2*z + 1) * q^48 - 2*z * q^49 + (4*z - 8) * q^51 + (-2*z + 2) * q^52 + 2 * q^53 + (-3*z - 3) * q^54 + (9*z - 9) * q^56 + (8*z - 16) * q^57 - z * q^58 + 14*z * q^59 + (7*z - 7) * q^61 + 9*z * q^63 + 7 * q^64 + (-2*z - 2) * q^66 - 3*z * q^67 - 4*z * q^68 + (3*z + 3) * q^69 + 2 * q^71 + (-9*z + 9) * q^72 - 4 * q^73 + (-4*z + 4) * q^74 - 8*z * q^76 + 6*z * q^77 + (2*z - 4) * q^78 + (-6*z + 6) * q^79 - 9*z * q^81 - 5 * q^82 + (-9*z + 9) * q^83 + (3*z - 6) * q^84 + 8*z * q^86 + (-2*z + 1) * q^87 + (-6*z + 6) * q^88 - 15 * q^89 + 6 * q^91 + (3*z - 3) * q^92 - 7*z * q^94 + (5*z + 5) * q^96 + (-2*z + 2) * q^97 - 2 * q^98 - 6*z * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} + 3 q^{3} + q^{4} - 3 q^{7} + 6 q^{8} + 3 q^{9}+O(q^{10})$$ 2 * q + q^2 + 3 * q^3 + q^4 - 3 * q^7 + 6 * q^8 + 3 * q^9 $$2 q + q^{2} + 3 q^{3} + q^{4} - 3 q^{7} + 6 q^{8} + 3 q^{9} + 2 q^{11} + 3 q^{12} - 2 q^{13} + 3 q^{14} + q^{16} - 8 q^{17} - 3 q^{18} - 16 q^{19} - 2 q^{22} + 3 q^{23} + 9 q^{24} - 4 q^{26} - 6 q^{28} + q^{29} + 5 q^{32} - 4 q^{34} + 6 q^{36} + 8 q^{37} - 8 q^{38} - 6 q^{39} - 5 q^{41} + 9 q^{42} - 8 q^{43} + 4 q^{44} + 6 q^{46} + 7 q^{47} - 2 q^{49} - 12 q^{51} + 2 q^{52} + 4 q^{53} - 9 q^{54} - 9 q^{56} - 24 q^{57} - q^{58} + 14 q^{59} - 7 q^{61} + 9 q^{63} + 14 q^{64} - 6 q^{66} - 3 q^{67} - 4 q^{68} + 9 q^{69} + 4 q^{71} + 9 q^{72} - 8 q^{73} + 4 q^{74} - 8 q^{76} + 6 q^{77} - 6 q^{78} + 6 q^{79} - 9 q^{81} - 10 q^{82} + 9 q^{83} - 9 q^{84} + 8 q^{86} + 6 q^{88} - 30 q^{89} + 12 q^{91} - 3 q^{92} - 7 q^{94} + 15 q^{96} + 2 q^{97} - 4 q^{98} - 6 q^{99}+O(q^{100})$$ 2 * q + q^2 + 3 * q^3 + q^4 - 3 * q^7 + 6 * q^8 + 3 * q^9 + 2 * q^11 + 3 * q^12 - 2 * q^13 + 3 * q^14 + q^16 - 8 * q^17 - 3 * q^18 - 16 * q^19 - 2 * q^22 + 3 * q^23 + 9 * q^24 - 4 * q^26 - 6 * q^28 + q^29 + 5 * q^32 - 4 * q^34 + 6 * q^36 + 8 * q^37 - 8 * q^38 - 6 * q^39 - 5 * q^41 + 9 * q^42 - 8 * q^43 + 4 * q^44 + 6 * q^46 + 7 * q^47 - 2 * q^49 - 12 * q^51 + 2 * q^52 + 4 * q^53 - 9 * q^54 - 9 * q^56 - 24 * q^57 - q^58 + 14 * q^59 - 7 * q^61 + 9 * q^63 + 14 * q^64 - 6 * q^66 - 3 * q^67 - 4 * q^68 + 9 * q^69 + 4 * q^71 + 9 * q^72 - 8 * q^73 + 4 * q^74 - 8 * q^76 + 6 * q^77 - 6 * q^78 + 6 * q^79 - 9 * q^81 - 10 * q^82 + 9 * q^83 - 9 * q^84 + 8 * q^86 + 6 * q^88 - 30 * q^89 + 12 * q^91 - 3 * q^92 - 7 * q^94 + 15 * q^96 + 2 * q^97 - 4 * q^98 - 6 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/225\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
76.1
 0.5 − 0.866025i 0.5 + 0.866025i
0.500000 + 0.866025i 1.50000 + 0.866025i 0.500000 0.866025i 0 1.73205i −1.50000 2.59808i 3.00000 1.50000 + 2.59808i 0
151.1 0.500000 0.866025i 1.50000 0.866025i 0.500000 + 0.866025i 0 1.73205i −1.50000 + 2.59808i 3.00000 1.50000 2.59808i 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.2.e.a 2
3.b odd 2 1 675.2.e.a 2
5.b even 2 1 45.2.e.a 2
5.c odd 4 2 225.2.k.a 4
9.c even 3 1 inner 225.2.e.a 2
9.c even 3 1 2025.2.a.b 1
9.d odd 6 1 675.2.e.a 2
9.d odd 6 1 2025.2.a.e 1
15.d odd 2 1 135.2.e.a 2
15.e even 4 2 675.2.k.a 4
20.d odd 2 1 720.2.q.d 2
45.h odd 6 1 135.2.e.a 2
45.h odd 6 1 405.2.a.b 1
45.j even 6 1 45.2.e.a 2
45.j even 6 1 405.2.a.e 1
45.k odd 12 2 225.2.k.a 4
45.k odd 12 2 2025.2.b.c 2
45.l even 12 2 675.2.k.a 4
45.l even 12 2 2025.2.b.d 2
60.h even 2 1 2160.2.q.a 2
180.n even 6 1 2160.2.q.a 2
180.n even 6 1 6480.2.a.x 1
180.p odd 6 1 720.2.q.d 2
180.p odd 6 1 6480.2.a.k 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.2.e.a 2 5.b even 2 1
45.2.e.a 2 45.j even 6 1
135.2.e.a 2 15.d odd 2 1
135.2.e.a 2 45.h odd 6 1
225.2.e.a 2 1.a even 1 1 trivial
225.2.e.a 2 9.c even 3 1 inner
225.2.k.a 4 5.c odd 4 2
225.2.k.a 4 45.k odd 12 2
405.2.a.b 1 45.h odd 6 1
405.2.a.e 1 45.j even 6 1
675.2.e.a 2 3.b odd 2 1
675.2.e.a 2 9.d odd 6 1
675.2.k.a 4 15.e even 4 2
675.2.k.a 4 45.l even 12 2
720.2.q.d 2 20.d odd 2 1
720.2.q.d 2 180.p odd 6 1
2025.2.a.b 1 9.c even 3 1
2025.2.a.e 1 9.d odd 6 1
2025.2.b.c 2 45.k odd 12 2
2025.2.b.d 2 45.l even 12 2
2160.2.q.a 2 60.h even 2 1
2160.2.q.a 2 180.n even 6 1
6480.2.a.k 1 180.p odd 6 1
6480.2.a.x 1 180.n even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(225, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T + 1$$
$3$ $$T^{2} - 3T + 3$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 3T + 9$$
$11$ $$T^{2} - 2T + 4$$
$13$ $$T^{2} + 2T + 4$$
$17$ $$(T + 4)^{2}$$
$19$ $$(T + 8)^{2}$$
$23$ $$T^{2} - 3T + 9$$
$29$ $$T^{2} - T + 1$$
$31$ $$T^{2}$$
$37$ $$(T - 4)^{2}$$
$41$ $$T^{2} + 5T + 25$$
$43$ $$T^{2} + 8T + 64$$
$47$ $$T^{2} - 7T + 49$$
$53$ $$(T - 2)^{2}$$
$59$ $$T^{2} - 14T + 196$$
$61$ $$T^{2} + 7T + 49$$
$67$ $$T^{2} + 3T + 9$$
$71$ $$(T - 2)^{2}$$
$73$ $$(T + 4)^{2}$$
$79$ $$T^{2} - 6T + 36$$
$83$ $$T^{2} - 9T + 81$$
$89$ $$(T + 15)^{2}$$
$97$ $$T^{2} - 2T + 4$$