# Properties

 Label 225.2.h.b Level $225$ Weight $2$ Character orbit 225.h Analytic conductor $1.797$ Analytic rank $0$ 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] = [225,2,Mod(46,225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(225, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([0, 6]))

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

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

chi := DirichletCharacter("225.46");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$225 = 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 225.h (of order $$5$$, degree $$4$$, 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: $$4$$ Coefficient field: $$\Q(\zeta_{10})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} + x^{2} - x + 1$$ x^4 - x^3 + x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 25) Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-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 $$\zeta_{10}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{10}^{2} + \zeta_{10}) q^{2} + (\zeta_{10}^{3} + \zeta_{10} - 1) q^{4} + (2 \zeta_{10}^{2} - \zeta_{10} + 2) q^{5} + (\zeta_{10}^{3} - \zeta_{10}^{2} - 1) q^{7} + (2 \zeta_{10}^{3} - \zeta_{10}^{2} + 2 \zeta_{10}) q^{8}+O(q^{10})$$ q + (-z^2 + z) * q^2 + (z^3 + z - 1) * q^4 + (2*z^2 - z + 2) * q^5 + (z^3 - z^2 - 1) * q^7 + (2*z^3 - z^2 + 2*z) * q^8 $$q + ( - \zeta_{10}^{2} + \zeta_{10}) q^{2} + (\zeta_{10}^{3} + \zeta_{10} - 1) q^{4} + (2 \zeta_{10}^{2} - \zeta_{10} + 2) q^{5} + (\zeta_{10}^{3} - \zeta_{10}^{2} - 1) q^{7} + (2 \zeta_{10}^{3} - \zeta_{10}^{2} + 2 \zeta_{10}) q^{8} + (\zeta_{10}^{3} - \zeta_{10}^{2} + 2) q^{10} + (2 \zeta_{10}^{3} - 4 \zeta_{10}^{2} + 4 \zeta_{10} - 2) q^{11} + (3 \zeta_{10}^{2} + 3) q^{13} + (\zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 1) q^{14} + ( - 3 \zeta_{10}^{2} + 3 \zeta_{10} - 3) q^{16} + ( - 2 \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 2 \zeta_{10}) q^{17} + ( - 3 \zeta_{10}^{3} - \zeta_{10}^{2} - 3 \zeta_{10}) q^{19} + (3 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10} - 3) q^{20} + ( - 2 \zeta_{10}^{3} + 4 \zeta_{10} - 4) q^{22} + ( - 7 \zeta_{10}^{3} + 5 \zeta_{10}^{2} - 5 \zeta_{10} + 7) q^{23} + 5 \zeta_{10}^{2} q^{25} + 3 q^{26} + ( - 2 \zeta_{10}^{3} - \zeta_{10} + 1) q^{28} + ( - 2 \zeta_{10}^{3} + \zeta_{10} - 1) q^{29} - 3 \zeta_{10}^{2} q^{31} + (\zeta_{10}^{3} - \zeta_{10}^{2} - 5) q^{32} + (4 \zeta_{10}^{2} - 6 \zeta_{10} + 4) q^{34} + ( - \zeta_{10}^{2} - 2 \zeta_{10} - 1) q^{35} + ( - 2 \zeta_{10}^{2} - \zeta_{10} - 2) q^{37} + ( - \zeta_{10}^{2} - 2 \zeta_{10} - 1) q^{38} + 5 \zeta_{10}^{3} q^{40} + ( - 2 \zeta_{10}^{2} - 2 \zeta_{10} - 2) q^{41} + ( - 3 \zeta_{10}^{3} + 3 \zeta_{10}^{2}) q^{43} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10}) q^{44} + ( - 2 \zeta_{10}^{3} - 5 \zeta_{10} + 5) q^{46} + ( - \zeta_{10}^{3} - \zeta_{10} + 1) q^{47} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - 5) q^{49} + (5 \zeta_{10}^{2} - 5 \zeta_{10} + 5) q^{50} + (6 \zeta_{10}^{3} - 3 \zeta_{10}^{2} + 3 \zeta_{10} - 6) q^{52} + (3 \zeta_{10}^{3} + 4 \zeta_{10} - 4) q^{53} + (6 \zeta_{10}^{3} - 6 \zeta_{10}^{2} + 2) q^{55} + ( - \zeta_{10}^{3} - 2 \zeta_{10}^{2} - \zeta_{10}) q^{56} + ( - 3 \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 3 \zeta_{10}) q^{58} + (3 \zeta_{10}^{2} - 9 \zeta_{10} + 3) q^{59} + ( - \zeta_{10}^{3} - 5 \zeta_{10}^{2} + 5 \zeta_{10} + 1) q^{61} + ( - 3 \zeta_{10}^{2} + 3 \zeta_{10} - 3) q^{62} + (\zeta_{10}^{3} - 3 \zeta_{10}^{2} + 3 \zeta_{10} - 1) q^{64} + (3 \zeta_{10}^{3} + 6 \zeta_{10}^{2} + 3 \zeta_{10}) q^{65} + (2 \zeta_{10}^{3} + 6 \zeta_{10}^{2} + 2 \zeta_{10}) q^{67} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2}) q^{68} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} - 1) q^{70} + ( - 5 \zeta_{10}^{3} + \zeta_{10} - 1) q^{71} + ( - 9 \zeta_{10}^{3} + 9 \zeta_{10}^{2} - 9 \zeta_{10} + 9) q^{73} + (\zeta_{10}^{3} - \zeta_{10}^{2} - 2) q^{74} + ( - 4 \zeta_{10}^{3} + 4 \zeta_{10}^{2} + 7) q^{76} + ( - 2 \zeta_{10}^{2} + 2 \zeta_{10}) q^{77} + ( - 5 \zeta_{10} + 5) q^{79} + (3 \zeta_{10}^{3} - 9 \zeta_{10}^{2} + 3 \zeta_{10}) q^{80} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} - 2) q^{82} + ( - 2 \zeta_{10}^{3} + 5 \zeta_{10}^{2} - 2 \zeta_{10}) q^{83} + ( - 2 \zeta_{10}^{3} + 6 \zeta_{10} - 6) q^{85} + ( - 3 \zeta_{10}^{3} + 6 \zeta_{10}^{2} - 6 \zeta_{10} + 3) q^{86} + ( - 6 \zeta_{10}^{2} + 8 \zeta_{10} - 6) q^{88} + ( - 4 \zeta_{10}^{3} - 4 \zeta_{10}^{2} + 4 \zeta_{10} + 4) q^{89} + ( - 3 \zeta_{10}^{2} - 3 \zeta_{10} - 3) q^{91} + (7 \zeta_{10}^{3} + 2 \zeta_{10}^{2} + 7 \zeta_{10}) q^{92} - \zeta_{10}^{2} q^{94} + ( - 10 \zeta_{10}^{3} - 5 \zeta_{10} + 5) q^{95} + ( - \zeta_{10}^{3} - 3 \zeta_{10} + 3) q^{97} + ( - \zeta_{10}^{3} + 7 \zeta_{10}^{2} - 7 \zeta_{10} + 1) q^{98} +O(q^{100})$$ q + (-z^2 + z) * q^2 + (z^3 + z - 1) * q^4 + (2*z^2 - z + 2) * q^5 + (z^3 - z^2 - 1) * q^7 + (2*z^3 - z^2 + 2*z) * q^8 + (z^3 - z^2 + 2) * q^10 + (2*z^3 - 4*z^2 + 4*z - 2) * q^11 + (3*z^2 + 3) * q^13 + (z^3 - z^2 + z - 1) * q^14 + (-3*z^2 + 3*z - 3) * q^16 + (-2*z^3 + 4*z^2 - 2*z) * q^17 + (-3*z^3 - z^2 - 3*z) * q^19 + (3*z^3 - 2*z^2 + 2*z - 3) * q^20 + (-2*z^3 + 4*z - 4) * q^22 + (-7*z^3 + 5*z^2 - 5*z + 7) * q^23 + 5*z^2 * q^25 + 3 * q^26 + (-2*z^3 - z + 1) * q^28 + (-2*z^3 + z - 1) * q^29 - 3*z^2 * q^31 + (z^3 - z^2 - 5) * q^32 + (4*z^2 - 6*z + 4) * q^34 + (-z^2 - 2*z - 1) * q^35 + (-2*z^2 - z - 2) * q^37 + (-z^2 - 2*z - 1) * q^38 + 5*z^3 * q^40 + (-2*z^2 - 2*z - 2) * q^41 + (-3*z^3 + 3*z^2) * q^43 + (-2*z^3 + 2*z^2 - 2*z) * q^44 + (-2*z^3 - 5*z + 5) * q^46 + (-z^3 - z + 1) * q^47 + (-z^3 + z^2 - 5) * q^49 + (5*z^2 - 5*z + 5) * q^50 + (6*z^3 - 3*z^2 + 3*z - 6) * q^52 + (3*z^3 + 4*z - 4) * q^53 + (6*z^3 - 6*z^2 + 2) * q^55 + (-z^3 - 2*z^2 - z) * q^56 + (-3*z^3 + 4*z^2 - 3*z) * q^58 + (3*z^2 - 9*z + 3) * q^59 + (-z^3 - 5*z^2 + 5*z + 1) * q^61 + (-3*z^2 + 3*z - 3) * q^62 + (z^3 - 3*z^2 + 3*z - 1) * q^64 + (3*z^3 + 6*z^2 + 3*z) * q^65 + (2*z^3 + 6*z^2 + 2*z) * q^67 + (2*z^3 - 2*z^2) * q^68 + (2*z^3 - 2*z^2 - 1) * q^70 + (-5*z^3 + z - 1) * q^71 + (-9*z^3 + 9*z^2 - 9*z + 9) * q^73 + (z^3 - z^2 - 2) * q^74 + (-4*z^3 + 4*z^2 + 7) * q^76 + (-2*z^2 + 2*z) * q^77 + (-5*z + 5) * q^79 + (3*z^3 - 9*z^2 + 3*z) * q^80 + (2*z^3 - 2*z^2 - 2) * q^82 + (-2*z^3 + 5*z^2 - 2*z) * q^83 + (-2*z^3 + 6*z - 6) * q^85 + (-3*z^3 + 6*z^2 - 6*z + 3) * q^86 + (-6*z^2 + 8*z - 6) * q^88 + (-4*z^3 - 4*z^2 + 4*z + 4) * q^89 + (-3*z^2 - 3*z - 3) * q^91 + (7*z^3 + 2*z^2 + 7*z) * q^92 - z^2 * q^94 + (-10*z^3 - 5*z + 5) * q^95 + (-z^3 - 3*z + 3) * q^97 + (-z^3 + 7*z^2 - 7*z + 1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{2} - 2 q^{4} + 5 q^{5} - 2 q^{7} + 5 q^{8}+O(q^{10})$$ 4 * q + 2 * q^2 - 2 * q^4 + 5 * q^5 - 2 * q^7 + 5 * q^8 $$4 q + 2 q^{2} - 2 q^{4} + 5 q^{5} - 2 q^{7} + 5 q^{8} + 10 q^{10} + 2 q^{11} + 9 q^{13} - q^{14} - 6 q^{16} - 8 q^{17} - 5 q^{19} - 5 q^{20} - 14 q^{22} + 11 q^{23} - 5 q^{25} + 12 q^{26} + q^{28} - 5 q^{29} + 3 q^{31} - 18 q^{32} + 6 q^{34} - 5 q^{35} - 7 q^{37} - 5 q^{38} + 5 q^{40} - 8 q^{41} - 6 q^{43} - 6 q^{44} + 13 q^{46} + 2 q^{47} - 22 q^{49} + 10 q^{50} - 12 q^{52} - 9 q^{53} + 20 q^{55} - 10 q^{58} + 13 q^{61} - 6 q^{62} + 3 q^{64} - 2 q^{67} + 4 q^{68} - 8 q^{71} + 9 q^{73} - 6 q^{74} + 20 q^{76} + 4 q^{77} + 15 q^{79} + 15 q^{80} - 4 q^{82} - 9 q^{83} - 20 q^{85} - 3 q^{86} - 10 q^{88} + 20 q^{89} - 12 q^{91} + 12 q^{92} + q^{94} + 5 q^{95} + 8 q^{97} - 11 q^{98}+O(q^{100})$$ 4 * q + 2 * q^2 - 2 * q^4 + 5 * q^5 - 2 * q^7 + 5 * q^8 + 10 * q^10 + 2 * q^11 + 9 * q^13 - q^14 - 6 * q^16 - 8 * q^17 - 5 * q^19 - 5 * q^20 - 14 * q^22 + 11 * q^23 - 5 * q^25 + 12 * q^26 + q^28 - 5 * q^29 + 3 * q^31 - 18 * q^32 + 6 * q^34 - 5 * q^35 - 7 * q^37 - 5 * q^38 + 5 * q^40 - 8 * q^41 - 6 * q^43 - 6 * q^44 + 13 * q^46 + 2 * q^47 - 22 * q^49 + 10 * q^50 - 12 * q^52 - 9 * q^53 + 20 * q^55 - 10 * q^58 + 13 * q^61 - 6 * q^62 + 3 * q^64 - 2 * q^67 + 4 * q^68 - 8 * q^71 + 9 * q^73 - 6 * q^74 + 20 * q^76 + 4 * q^77 + 15 * q^79 + 15 * q^80 - 4 * q^82 - 9 * q^83 - 20 * q^85 - 3 * q^86 - 10 * q^88 + 20 * q^89 - 12 * q^91 + 12 * q^92 + q^94 + 5 * q^95 + 8 * q^97 - 11 * q^98

## Character values

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

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$1$$ $$-\zeta_{10}^{3}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
46.1
 −0.309017 − 0.951057i 0.809017 − 0.587785i 0.809017 + 0.587785i −0.309017 + 0.951057i
0.500000 1.53884i 0 −0.500000 0.363271i 0.690983 + 2.12663i 0 0.618034 1.80902 1.31433i 0 3.61803
91.1 0.500000 + 0.363271i 0 −0.500000 1.53884i 1.80902 1.31433i 0 −1.61803 0.690983 2.12663i 0 1.38197
136.1 0.500000 0.363271i 0 −0.500000 + 1.53884i 1.80902 + 1.31433i 0 −1.61803 0.690983 + 2.12663i 0 1.38197
181.1 0.500000 + 1.53884i 0 −0.500000 + 0.363271i 0.690983 2.12663i 0 0.618034 1.80902 + 1.31433i 0 3.61803
 $$n$$: 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
25.d even 5 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.2.h.b 4
3.b odd 2 1 25.2.d.a 4
12.b even 2 1 400.2.u.b 4
15.d odd 2 1 125.2.d.a 4
15.e even 4 2 125.2.e.a 8
25.d even 5 1 inner 225.2.h.b 4
25.d even 5 1 5625.2.a.f 2
25.e even 10 1 5625.2.a.d 2
75.h odd 10 1 125.2.d.a 4
75.h odd 10 1 625.2.a.c 2
75.h odd 10 2 625.2.d.b 4
75.j odd 10 1 25.2.d.a 4
75.j odd 10 1 625.2.a.b 2
75.j odd 10 2 625.2.d.h 4
75.l even 20 2 125.2.e.a 8
75.l even 20 2 625.2.b.a 4
75.l even 20 4 625.2.e.c 8
300.n even 10 1 400.2.u.b 4
300.n even 10 1 10000.2.a.c 2
300.r even 10 1 10000.2.a.l 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.2.d.a 4 3.b odd 2 1
25.2.d.a 4 75.j odd 10 1
125.2.d.a 4 15.d odd 2 1
125.2.d.a 4 75.h odd 10 1
125.2.e.a 8 15.e even 4 2
125.2.e.a 8 75.l even 20 2
225.2.h.b 4 1.a even 1 1 trivial
225.2.h.b 4 25.d even 5 1 inner
400.2.u.b 4 12.b even 2 1
400.2.u.b 4 300.n even 10 1
625.2.a.b 2 75.j odd 10 1
625.2.a.c 2 75.h odd 10 1
625.2.b.a 4 75.l even 20 2
625.2.d.b 4 75.h odd 10 2
625.2.d.h 4 75.j odd 10 2
625.2.e.c 8 75.l even 20 4
5625.2.a.d 2 25.e even 10 1
5625.2.a.f 2 25.d even 5 1
10000.2.a.c 2 300.n even 10 1
10000.2.a.l 2 300.r even 10 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} - 2T_{2}^{3} + 4T_{2}^{2} - 3T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(225, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 2 T^{3} + 4 T^{2} - 3 T + 1$$
$3$ $$T^{4}$$
$5$ $$T^{4} - 5 T^{3} + 15 T^{2} - 25 T + 25$$
$7$ $$(T^{2} + T - 1)^{2}$$
$11$ $$T^{4} - 2 T^{3} + 24 T^{2} + 32 T + 16$$
$13$ $$T^{4} - 9 T^{3} + 36 T^{2} - 54 T + 81$$
$17$ $$T^{4} + 8 T^{3} + 24 T^{2} - 8 T + 16$$
$19$ $$T^{4} + 5 T^{3} + 40 T^{2} + 50 T + 25$$
$23$ $$T^{4} - 11 T^{3} + 51 T^{2} + \cdots + 961$$
$29$ $$T^{4} + 5 T^{3} + 10 T^{2} + 25$$
$31$ $$T^{4} - 3 T^{3} + 9 T^{2} - 27 T + 81$$
$37$ $$T^{4} + 7 T^{3} + 19 T^{2} + 3 T + 1$$
$41$ $$T^{4} + 8 T^{3} + 24 T^{2} - 8 T + 16$$
$43$ $$(T^{2} + 3 T - 9)^{2}$$
$47$ $$T^{4} - 2 T^{3} + 4 T^{2} - 3 T + 1$$
$53$ $$T^{4} + 9 T^{3} + 61 T^{2} + 209 T + 361$$
$59$ $$T^{4} + 90 T^{2} + 675 T + 2025$$
$61$ $$T^{4} - 13 T^{3} + 139 T^{2} + \cdots + 1681$$
$67$ $$T^{4} + 2 T^{3} + 64 T^{2} + \cdots + 1936$$
$71$ $$T^{4} + 8 T^{3} + 34 T^{2} + 87 T + 841$$
$73$ $$T^{4} - 9 T^{3} + 81 T^{2} + \cdots + 6561$$
$79$ $$T^{4} - 15 T^{3} + 100 T^{2} + \cdots + 625$$
$83$ $$T^{4} + 9 T^{3} + 31 T^{2} - 11 T + 121$$
$89$ $$T^{4} - 20 T^{3} + 240 T^{2} + \cdots + 6400$$
$97$ $$T^{4} - 8 T^{3} + 34 T^{2} - 77 T + 121$$