# Properties

 Label 225.2.k.a Level $225$ Weight $2$ Character orbit 225.k Analytic conductor $1.797$ Analytic rank $0$ Dimension $4$ Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,2,Mod(49,225)]

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, 3]))

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

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

chi := DirichletCharacter("225.49");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$225 = 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 225.k (of order $$6$$, degree $$2$$, not 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 45) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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 $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \zeta_{12} q^{2} + ( - \zeta_{12}^{3} - \zeta_{12}) q^{3} - \zeta_{12}^{2} q^{4} + ( - 2 \zeta_{12}^{2} + 1) q^{6} - 3 \zeta_{12} q^{7} - 3 \zeta_{12}^{3} q^{8} + (3 \zeta_{12}^{2} - 3) q^{9} +O(q^{10})$$ q + z * q^2 + (-z^3 - z) * q^3 - z^2 * q^4 + (-2*z^2 + 1) * q^6 - 3*z * q^7 - 3*z^3 * q^8 + (3*z^2 - 3) * q^9 $$q + \zeta_{12} q^{2} + ( - \zeta_{12}^{3} - \zeta_{12}) q^{3} - \zeta_{12}^{2} q^{4} + ( - 2 \zeta_{12}^{2} + 1) q^{6} - 3 \zeta_{12} q^{7} - 3 \zeta_{12}^{3} q^{8} + (3 \zeta_{12}^{2} - 3) q^{9} + ( - 2 \zeta_{12}^{2} + 2) q^{11} + (2 \zeta_{12}^{3} - \zeta_{12}) q^{12} + (2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{13} - 3 \zeta_{12}^{2} q^{14} + ( - \zeta_{12}^{2} + 1) q^{16} - 4 \zeta_{12}^{3} q^{17} + (3 \zeta_{12}^{3} - 3 \zeta_{12}) q^{18} + 8 q^{19} + (6 \zeta_{12}^{2} - 3) q^{21} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{22} + ( - 3 \zeta_{12}^{3} + 3 \zeta_{12}) q^{23} + (3 \zeta_{12}^{2} - 6) q^{24} - 2 q^{26} + ( - 3 \zeta_{12}^{3} + 6 \zeta_{12}) q^{27} + 3 \zeta_{12}^{3} q^{28} + (\zeta_{12}^{2} - 1) q^{29} + (5 \zeta_{12}^{3} - 5 \zeta_{12}) q^{32} + (2 \zeta_{12}^{3} - 4 \zeta_{12}) q^{33} + ( - 4 \zeta_{12}^{2} + 4) q^{34} + 3 q^{36} + 4 \zeta_{12}^{3} q^{37} + 8 \zeta_{12} q^{38} + (2 \zeta_{12}^{2} + 2) q^{39} - 5 \zeta_{12}^{2} q^{41} + (6 \zeta_{12}^{3} - 3 \zeta_{12}) q^{42} + 8 \zeta_{12} q^{43} - 2 q^{44} + 3 q^{46} + 7 \zeta_{12} q^{47} + (\zeta_{12}^{3} - 2 \zeta_{12}) q^{48} + 2 \zeta_{12}^{2} q^{49} + (4 \zeta_{12}^{2} - 8) q^{51} + 2 \zeta_{12} q^{52} - 2 \zeta_{12}^{3} q^{53} + (3 \zeta_{12}^{2} + 3) q^{54} + (9 \zeta_{12}^{2} - 9) q^{56} + ( - 8 \zeta_{12}^{3} - 8 \zeta_{12}) q^{57} + (\zeta_{12}^{3} - \zeta_{12}) q^{58} - 14 \zeta_{12}^{2} q^{59} + (7 \zeta_{12}^{2} - 7) q^{61} + ( - 9 \zeta_{12}^{3} + 9 \zeta_{12}) q^{63} - 7 q^{64} + ( - 2 \zeta_{12}^{2} - 2) q^{66} + ( - 3 \zeta_{12}^{3} + 3 \zeta_{12}) q^{67} + (4 \zeta_{12}^{3} - 4 \zeta_{12}) q^{68} + ( - 3 \zeta_{12}^{2} - 3) q^{69} + 2 q^{71} + 9 \zeta_{12} q^{72} + 4 \zeta_{12}^{3} q^{73} + (4 \zeta_{12}^{2} - 4) q^{74} - 8 \zeta_{12}^{2} q^{76} + (6 \zeta_{12}^{3} - 6 \zeta_{12}) q^{77} + (2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{78} + (6 \zeta_{12}^{2} - 6) q^{79} - 9 \zeta_{12}^{2} q^{81} - 5 \zeta_{12}^{3} q^{82} - 9 \zeta_{12} q^{83} + ( - 3 \zeta_{12}^{2} + 6) q^{84} + 8 \zeta_{12}^{2} q^{86} + ( - \zeta_{12}^{3} + 2 \zeta_{12}) q^{87} - 6 \zeta_{12} q^{88} + 15 q^{89} + 6 q^{91} - 3 \zeta_{12} q^{92} + 7 \zeta_{12}^{2} q^{94} + (5 \zeta_{12}^{2} + 5) q^{96} + 2 \zeta_{12} q^{97} + 2 \zeta_{12}^{3} q^{98} + 6 \zeta_{12}^{2} q^{99} +O(q^{100})$$ q + z * q^2 + (-z^3 - z) * q^3 - z^2 * q^4 + (-2*z^2 + 1) * q^6 - 3*z * q^7 - 3*z^3 * q^8 + (3*z^2 - 3) * q^9 + (-2*z^2 + 2) * q^11 + (2*z^3 - z) * q^12 + (2*z^3 - 2*z) * q^13 - 3*z^2 * q^14 + (-z^2 + 1) * q^16 - 4*z^3 * q^17 + (3*z^3 - 3*z) * q^18 + 8 * q^19 + (6*z^2 - 3) * q^21 + (-2*z^3 + 2*z) * q^22 + (-3*z^3 + 3*z) * q^23 + (3*z^2 - 6) * q^24 - 2 * q^26 + (-3*z^3 + 6*z) * q^27 + 3*z^3 * q^28 + (z^2 - 1) * q^29 + (5*z^3 - 5*z) * q^32 + (2*z^3 - 4*z) * q^33 + (-4*z^2 + 4) * q^34 + 3 * q^36 + 4*z^3 * q^37 + 8*z * q^38 + (2*z^2 + 2) * q^39 - 5*z^2 * q^41 + (6*z^3 - 3*z) * q^42 + 8*z * q^43 - 2 * q^44 + 3 * q^46 + 7*z * q^47 + (z^3 - 2*z) * q^48 + 2*z^2 * q^49 + (4*z^2 - 8) * q^51 + 2*z * q^52 - 2*z^3 * q^53 + (3*z^2 + 3) * q^54 + (9*z^2 - 9) * q^56 + (-8*z^3 - 8*z) * q^57 + (z^3 - z) * q^58 - 14*z^2 * q^59 + (7*z^2 - 7) * q^61 + (-9*z^3 + 9*z) * q^63 - 7 * q^64 + (-2*z^2 - 2) * q^66 + (-3*z^3 + 3*z) * q^67 + (4*z^3 - 4*z) * q^68 + (-3*z^2 - 3) * q^69 + 2 * q^71 + 9*z * q^72 + 4*z^3 * q^73 + (4*z^2 - 4) * q^74 - 8*z^2 * q^76 + (6*z^3 - 6*z) * q^77 + (2*z^3 + 2*z) * q^78 + (6*z^2 - 6) * q^79 - 9*z^2 * q^81 - 5*z^3 * q^82 - 9*z * q^83 + (-3*z^2 + 6) * q^84 + 8*z^2 * q^86 + (-z^3 + 2*z) * q^87 - 6*z * q^88 + 15 * q^89 + 6 * q^91 - 3*z * q^92 + 7*z^2 * q^94 + (5*z^2 + 5) * q^96 + 2*z * q^97 + 2*z^3 * q^98 + 6*z^2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{4} - 6 q^{9}+O(q^{10})$$ 4 * q - 2 * q^4 - 6 * q^9 $$4 q - 2 q^{4} - 6 q^{9} + 4 q^{11} - 6 q^{14} + 2 q^{16} + 32 q^{19} - 18 q^{24} - 8 q^{26} - 2 q^{29} + 8 q^{34} + 12 q^{36} + 12 q^{39} - 10 q^{41} - 8 q^{44} + 12 q^{46} + 4 q^{49} - 24 q^{51} + 18 q^{54} - 18 q^{56} - 28 q^{59} - 14 q^{61} - 28 q^{64} - 12 q^{66} - 18 q^{69} + 8 q^{71} - 8 q^{74} - 16 q^{76} - 12 q^{79} - 18 q^{81} + 18 q^{84} + 16 q^{86} + 60 q^{89} + 24 q^{91} + 14 q^{94} + 30 q^{96} + 12 q^{99}+O(q^{100})$$ 4 * q - 2 * q^4 - 6 * q^9 + 4 * q^11 - 6 * q^14 + 2 * q^16 + 32 * q^19 - 18 * q^24 - 8 * q^26 - 2 * q^29 + 8 * q^34 + 12 * q^36 + 12 * q^39 - 10 * q^41 - 8 * q^44 + 12 * q^46 + 4 * q^49 - 24 * q^51 + 18 * q^54 - 18 * q^56 - 28 * q^59 - 14 * q^61 - 28 * q^64 - 12 * q^66 - 18 * q^69 + 8 * q^71 - 8 * q^74 - 16 * q^76 - 12 * q^79 - 18 * q^81 + 18 * q^84 + 16 * q^86 + 60 * q^89 + 24 * q^91 + 14 * q^94 + 30 * q^96 + 12 * q^99

## Character values

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

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$-\zeta_{12}^{2}$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 −0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 − 0.500000i 0.866025 + 0.500000i
−0.866025 + 0.500000i 0.866025 1.50000i −0.500000 + 0.866025i 0 1.73205i 2.59808 1.50000i 3.00000i −1.50000 2.59808i 0
49.2 0.866025 0.500000i −0.866025 + 1.50000i −0.500000 + 0.866025i 0 1.73205i −2.59808 + 1.50000i 3.00000i −1.50000 2.59808i 0
124.1 −0.866025 0.500000i 0.866025 + 1.50000i −0.500000 0.866025i 0 1.73205i 2.59808 + 1.50000i 3.00000i −1.50000 + 2.59808i 0
124.2 0.866025 + 0.500000i −0.866025 1.50000i −0.500000 0.866025i 0 1.73205i −2.59808 1.50000i 3.00000i −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
5.b even 2 1 inner
9.c even 3 1 inner
45.j even 6 1 inner

## Twists

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

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} - T_{2}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(225, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - T^{2} + 1$$
$3$ $$T^{4} + 3T^{2} + 9$$
$5$ $$T^{4}$$
$7$ $$T^{4} - 9T^{2} + 81$$
$11$ $$(T^{2} - 2 T + 4)^{2}$$
$13$ $$T^{4} - 4T^{2} + 16$$
$17$ $$(T^{2} + 16)^{2}$$
$19$ $$(T - 8)^{4}$$
$23$ $$T^{4} - 9T^{2} + 81$$
$29$ $$(T^{2} + T + 1)^{2}$$
$31$ $$T^{4}$$
$37$ $$(T^{2} + 16)^{2}$$
$41$ $$(T^{2} + 5 T + 25)^{2}$$
$43$ $$T^{4} - 64T^{2} + 4096$$
$47$ $$T^{4} - 49T^{2} + 2401$$
$53$ $$(T^{2} + 4)^{2}$$
$59$ $$(T^{2} + 14 T + 196)^{2}$$
$61$ $$(T^{2} + 7 T + 49)^{2}$$
$67$ $$T^{4} - 9T^{2} + 81$$
$71$ $$(T - 2)^{4}$$
$73$ $$(T^{2} + 16)^{2}$$
$79$ $$(T^{2} + 6 T + 36)^{2}$$
$83$ $$T^{4} - 81T^{2} + 6561$$
$89$ $$(T - 15)^{4}$$
$97$ $$T^{4} - 4T^{2} + 16$$