# Properties

 Label 225.2.f.a Level $225$ Weight $2$ Character orbit 225.f Analytic conductor $1.797$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,2,Mod(107,225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(225, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 1]))

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

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

chi := DirichletCharacter("225.107");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$225 = 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 225.f (of order $$4$$, 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: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{8})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 45) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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 $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \zeta_{8} q^{2} - \zeta_{8}^{2} q^{4} + ( - 2 \zeta_{8}^{2} + 2) q^{7} - 3 \zeta_{8}^{3} q^{8} +O(q^{10})$$ q + z * q^2 - z^2 * q^4 + (-2*z^2 + 2) * q^7 - 3*z^3 * q^8 $$q + \zeta_{8} q^{2} - \zeta_{8}^{2} q^{4} + ( - 2 \zeta_{8}^{2} + 2) q^{7} - 3 \zeta_{8}^{3} q^{8} + (2 \zeta_{8}^{3} + 2 \zeta_{8}) q^{11} + ( - \zeta_{8}^{2} - 1) q^{13} + ( - 2 \zeta_{8}^{3} + 2 \zeta_{8}) q^{14} + q^{16} + 4 \zeta_{8} q^{17} + (2 \zeta_{8}^{2} - 2) q^{22} + 4 \zeta_{8}^{3} q^{23} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{26} + ( - 2 \zeta_{8}^{2} - 2) q^{28} + (3 \zeta_{8}^{3} - 3 \zeta_{8}) q^{29} - 4 q^{31} - 5 \zeta_{8} q^{32} + 4 \zeta_{8}^{2} q^{34} + (\zeta_{8}^{2} - 1) q^{37} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{41} + (8 \zeta_{8}^{2} + 8) q^{43} + ( - 2 \zeta_{8}^{3} + 2 \zeta_{8}) q^{44} - 4 q^{46} - 8 \zeta_{8} q^{47} - \zeta_{8}^{2} q^{49} + (\zeta_{8}^{2} - 1) q^{52} + 4 \zeta_{8}^{3} q^{53} + ( - 6 \zeta_{8}^{3} - 6 \zeta_{8}) q^{56} + ( - 3 \zeta_{8}^{2} - 3) q^{58} + (6 \zeta_{8}^{3} - 6 \zeta_{8}) q^{59} + 8 q^{61} - 4 \zeta_{8} q^{62} - 7 \zeta_{8}^{2} q^{64} + (4 \zeta_{8}^{2} - 4) q^{67} - 4 \zeta_{8}^{3} q^{68} + ( - 4 \zeta_{8}^{3} - 4 \zeta_{8}) q^{71} + ( - \zeta_{8}^{2} - 1) q^{73} + (\zeta_{8}^{3} - \zeta_{8}) q^{74} + 8 \zeta_{8} q^{77} + 12 \zeta_{8}^{2} q^{79} + ( - \zeta_{8}^{2} + 1) q^{82} + 4 \zeta_{8}^{3} q^{83} + (8 \zeta_{8}^{3} + 8 \zeta_{8}) q^{86} + (6 \zeta_{8}^{2} + 6) q^{88} + ( - 9 \zeta_{8}^{3} + 9 \zeta_{8}) q^{89} - 4 q^{91} + 4 \zeta_{8} q^{92} - 8 \zeta_{8}^{2} q^{94} + ( - 11 \zeta_{8}^{2} + 11) q^{97} - \zeta_{8}^{3} q^{98} +O(q^{100})$$ q + z * q^2 - z^2 * q^4 + (-2*z^2 + 2) * q^7 - 3*z^3 * q^8 + (2*z^3 + 2*z) * q^11 + (-z^2 - 1) * q^13 + (-2*z^3 + 2*z) * q^14 + q^16 + 4*z * q^17 + (2*z^2 - 2) * q^22 + 4*z^3 * q^23 + (-z^3 - z) * q^26 + (-2*z^2 - 2) * q^28 + (3*z^3 - 3*z) * q^29 - 4 * q^31 - 5*z * q^32 + 4*z^2 * q^34 + (z^2 - 1) * q^37 + (-z^3 - z) * q^41 + (8*z^2 + 8) * q^43 + (-2*z^3 + 2*z) * q^44 - 4 * q^46 - 8*z * q^47 - z^2 * q^49 + (z^2 - 1) * q^52 + 4*z^3 * q^53 + (-6*z^3 - 6*z) * q^56 + (-3*z^2 - 3) * q^58 + (6*z^3 - 6*z) * q^59 + 8 * q^61 - 4*z * q^62 - 7*z^2 * q^64 + (4*z^2 - 4) * q^67 - 4*z^3 * q^68 + (-4*z^3 - 4*z) * q^71 + (-z^2 - 1) * q^73 + (z^3 - z) * q^74 + 8*z * q^77 + 12*z^2 * q^79 + (-z^2 + 1) * q^82 + 4*z^3 * q^83 + (8*z^3 + 8*z) * q^86 + (6*z^2 + 6) * q^88 + (-9*z^3 + 9*z) * q^89 - 4 * q^91 + 4*z * q^92 - 8*z^2 * q^94 + (-11*z^2 + 11) * q^97 - z^3 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 8 q^{7}+O(q^{10})$$ 4 * q + 8 * q^7 $$4 q + 8 q^{7} - 4 q^{13} + 4 q^{16} - 8 q^{22} - 8 q^{28} - 16 q^{31} - 4 q^{37} + 32 q^{43} - 16 q^{46} - 4 q^{52} - 12 q^{58} + 32 q^{61} - 16 q^{67} - 4 q^{73} + 4 q^{82} + 24 q^{88} - 16 q^{91} + 44 q^{97}+O(q^{100})$$ 4 * q + 8 * q^7 - 4 * q^13 + 4 * q^16 - 8 * q^22 - 8 * q^28 - 16 * q^31 - 4 * q^37 + 32 * q^43 - 16 * q^46 - 4 * q^52 - 12 * q^58 + 32 * q^61 - 16 * q^67 - 4 * q^73 + 4 * q^82 + 24 * q^88 - 16 * q^91 + 44 * q^97

## Character values

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

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$-1$$ $$-\zeta_{8}^{2}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
107.1
 −0.707107 + 0.707107i 0.707107 − 0.707107i −0.707107 − 0.707107i 0.707107 + 0.707107i
−0.707107 + 0.707107i 0 1.00000i 0 0 2.00000 + 2.00000i −2.12132 2.12132i 0 0
107.2 0.707107 0.707107i 0 1.00000i 0 0 2.00000 + 2.00000i 2.12132 + 2.12132i 0 0
143.1 −0.707107 0.707107i 0 1.00000i 0 0 2.00000 2.00000i −2.12132 + 2.12132i 0 0
143.2 0.707107 + 0.707107i 0 1.00000i 0 0 2.00000 2.00000i 2.12132 2.12132i 0 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
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.2.f.a 4
3.b odd 2 1 inner 225.2.f.a 4
4.b odd 2 1 3600.2.w.b 4
5.b even 2 1 45.2.f.a 4
5.c odd 4 1 45.2.f.a 4
5.c odd 4 1 inner 225.2.f.a 4
12.b even 2 1 3600.2.w.b 4
15.d odd 2 1 45.2.f.a 4
15.e even 4 1 45.2.f.a 4
15.e even 4 1 inner 225.2.f.a 4
20.d odd 2 1 720.2.w.d 4
20.e even 4 1 720.2.w.d 4
20.e even 4 1 3600.2.w.b 4
40.e odd 2 1 2880.2.w.k 4
40.f even 2 1 2880.2.w.b 4
40.i odd 4 1 2880.2.w.b 4
40.k even 4 1 2880.2.w.k 4
45.h odd 6 2 405.2.m.a 8
45.j even 6 2 405.2.m.a 8
45.k odd 12 2 405.2.m.a 8
45.l even 12 2 405.2.m.a 8
60.h even 2 1 720.2.w.d 4
60.l odd 4 1 720.2.w.d 4
60.l odd 4 1 3600.2.w.b 4
120.i odd 2 1 2880.2.w.b 4
120.m even 2 1 2880.2.w.k 4
120.q odd 4 1 2880.2.w.k 4
120.w even 4 1 2880.2.w.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.2.f.a 4 5.b even 2 1
45.2.f.a 4 5.c odd 4 1
45.2.f.a 4 15.d odd 2 1
45.2.f.a 4 15.e even 4 1
225.2.f.a 4 1.a even 1 1 trivial
225.2.f.a 4 3.b odd 2 1 inner
225.2.f.a 4 5.c odd 4 1 inner
225.2.f.a 4 15.e even 4 1 inner
405.2.m.a 8 45.h odd 6 2
405.2.m.a 8 45.j even 6 2
405.2.m.a 8 45.k odd 12 2
405.2.m.a 8 45.l even 12 2
720.2.w.d 4 20.d odd 2 1
720.2.w.d 4 20.e even 4 1
720.2.w.d 4 60.h even 2 1
720.2.w.d 4 60.l odd 4 1
2880.2.w.b 4 40.f even 2 1
2880.2.w.b 4 40.i odd 4 1
2880.2.w.b 4 120.i odd 2 1
2880.2.w.b 4 120.w even 4 1
2880.2.w.k 4 40.e odd 2 1
2880.2.w.k 4 40.k even 4 1
2880.2.w.k 4 120.m even 2 1
2880.2.w.k 4 120.q odd 4 1
3600.2.w.b 4 4.b odd 2 1
3600.2.w.b 4 12.b even 2 1
3600.2.w.b 4 20.e even 4 1
3600.2.w.b 4 60.l odd 4 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 1$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$(T^{2} - 4 T + 8)^{2}$$
$11$ $$(T^{2} + 8)^{2}$$
$13$ $$(T^{2} + 2 T + 2)^{2}$$
$17$ $$T^{4} + 256$$
$19$ $$T^{4}$$
$23$ $$T^{4} + 256$$
$29$ $$(T^{2} - 18)^{2}$$
$31$ $$(T + 4)^{4}$$
$37$ $$(T^{2} + 2 T + 2)^{2}$$
$41$ $$(T^{2} + 2)^{2}$$
$43$ $$(T^{2} - 16 T + 128)^{2}$$
$47$ $$T^{4} + 4096$$
$53$ $$T^{4} + 256$$
$59$ $$(T^{2} - 72)^{2}$$
$61$ $$(T - 8)^{4}$$
$67$ $$(T^{2} + 8 T + 32)^{2}$$
$71$ $$(T^{2} + 32)^{2}$$
$73$ $$(T^{2} + 2 T + 2)^{2}$$
$79$ $$(T^{2} + 144)^{2}$$
$83$ $$T^{4} + 256$$
$89$ $$(T^{2} - 162)^{2}$$
$97$ $$(T^{2} - 22 T + 242)^{2}$$