# Properties

 Label 225.2.h.a 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 75) 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.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{2} + \zeta_{10}^{3} q^{4} + ( - 2 \zeta_{10}^{3} + \zeta_{10}^{2} - 2 \zeta_{10}) q^{5} + ( - 4 \zeta_{10}^{3} + 4 \zeta_{10}^{2} + 2) q^{7} + 3 \zeta_{10}^{2} q^{8}+O(q^{10})$$ q + (-z^3 + z^2 - z + 1) * q^2 + z^3 * q^4 + (-2*z^3 + z^2 - 2*z) * q^5 + (-4*z^3 + 4*z^2 + 2) * q^7 + 3*z^2 * q^8 $$q + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{2} + \zeta_{10}^{3} q^{4} + ( - 2 \zeta_{10}^{3} + \zeta_{10}^{2} - 2 \zeta_{10}) q^{5} + ( - 4 \zeta_{10}^{3} + 4 \zeta_{10}^{2} + 2) q^{7} + 3 \zeta_{10}^{2} q^{8} + ( - 2 \zeta_{10}^{2} + \zeta_{10} - 2) q^{10} + ( - 2 \zeta_{10}^{3} + 2) q^{11} + (\zeta_{10}^{2} - 5 \zeta_{10} + 1) q^{13} + ( - 2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10} + 2) q^{14} + \zeta_{10} q^{16} + (3 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 3 \zeta_{10}) q^{17} + ( - 2 \zeta_{10}^{3} - 2 \zeta_{10}) q^{19} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} + 1) q^{20} + ( - 2 \zeta_{10}^{3} - 2 \zeta_{10} + 2) q^{22} + (2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10} - 2) q^{23} + (5 \zeta_{10}^{3} - 5 \zeta_{10}^{2} + 5 \zeta_{10} - 5) q^{25} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - 4) q^{26} + (2 \zeta_{10}^{3} + 4 \zeta_{10} - 4) q^{28} + (5 \zeta_{10}^{3} - \zeta_{10} + 1) q^{29} + (2 \zeta_{10}^{3} + 4 \zeta_{10}^{2} + 2 \zeta_{10}) q^{31} - 5 q^{32} + (3 \zeta_{10}^{2} - 2 \zeta_{10} + 3) q^{34} - 10 \zeta_{10}^{2} q^{35} + ( - 5 \zeta_{10}^{2} - 5) q^{37} + ( - 2 \zeta_{10}^{2} - 2) q^{38} + ( - 3 \zeta_{10}^{3} - 3 \zeta_{10}^{2} + 3 \zeta_{10} + 3) q^{40} + (\zeta_{10}^{2} - 3 \zeta_{10} + 1) q^{41} + (6 \zeta_{10}^{3} - 6 \zeta_{10}^{2} - 2) q^{43} + (2 \zeta_{10}^{3} + 2 \zeta_{10}) q^{44} + (2 \zeta_{10}^{3} + 4 \zeta_{10} - 4) q^{46} + ( - 4 \zeta_{10}^{3} - 2 \zeta_{10} + 2) q^{47} + 13 q^{49} + 5 \zeta_{10}^{3} q^{50} + ( - 4 \zeta_{10}^{3} + 5 \zeta_{10}^{2} - 5 \zeta_{10} + 4) q^{52} + (2 \zeta_{10}^{3} - \zeta_{10} + 1) q^{53} + ( - 2 \zeta_{10}^{2} - 4 \zeta_{10} - 2) q^{55} + (12 \zeta_{10}^{3} - 6 \zeta_{10}^{2} + 12 \zeta_{10}) q^{56} + ( - \zeta_{10}^{3} + 6 \zeta_{10}^{2} - \zeta_{10}) q^{58} - 4 \zeta_{10} q^{59} + ( - \zeta_{10}^{2} + \zeta_{10}) q^{61} + (2 \zeta_{10}^{2} + 4 \zeta_{10} + 2) q^{62} + (7 \zeta_{10}^{3} - 7 \zeta_{10}^{2} + 7 \zeta_{10} - 7) q^{64} + (2 \zeta_{10}^{3} + 9 \zeta_{10} - 9) q^{65} + (2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} + 2 \zeta_{10}) q^{67} + (3 \zeta_{10}^{3} - 3 \zeta_{10}^{2} - 1) q^{68} - 10 \zeta_{10} q^{70} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10} - 2) q^{71} + (5 \zeta_{10}^{2} - 5 \zeta_{10}) q^{73} + (5 \zeta_{10}^{3} - 5 \zeta_{10}^{2} - 5) q^{74} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} + 2) q^{76} + ( - 12 \zeta_{10}^{3} + 8 \zeta_{10}^{2} - 8 \zeta_{10} + 12) q^{77} + ( - \zeta_{10}^{3} - 2 \zeta_{10} + 2) q^{80} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - 2) q^{82} + (4 \zeta_{10}^{3} - 10 \zeta_{10}^{2} + 4 \zeta_{10}) q^{83} + ( - 7 \zeta_{10}^{3} + 8 \zeta_{10}^{2} - 8 \zeta_{10} + 7) q^{85} + (2 \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 4 \zeta_{10} - 2) q^{86} + (6 \zeta_{10}^{2} + 6) q^{88} + ( - 7 \zeta_{10}^{3} + 6 \zeta_{10}^{2} - 6 \zeta_{10} + 7) q^{89} + ( - 18 \zeta_{10}^{2} + 14 \zeta_{10} - 18) q^{91} + ( - 4 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 4 \zeta_{10}) q^{92} + ( - 2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} - 2 \zeta_{10}) q^{94} + (6 \zeta_{10}^{3} - 4 \zeta_{10}^{2} + 4 \zeta_{10} - 6) q^{95} + ( - 7 \zeta_{10}^{3} - 3 \zeta_{10} + 3) q^{97} + ( - 13 \zeta_{10}^{3} + 13 \zeta_{10}^{2} - 13 \zeta_{10} + 13) q^{98} +O(q^{100})$$ q + (-z^3 + z^2 - z + 1) * q^2 + z^3 * q^4 + (-2*z^3 + z^2 - 2*z) * q^5 + (-4*z^3 + 4*z^2 + 2) * q^7 + 3*z^2 * q^8 + (-2*z^2 + z - 2) * q^10 + (-2*z^3 + 2) * q^11 + (z^2 - 5*z + 1) * q^13 + (-2*z^3 - 2*z^2 + 2*z + 2) * q^14 + z * q^16 + (3*z^3 - 2*z^2 + 3*z) * q^17 + (-2*z^3 - 2*z) * q^19 + (-2*z^3 + 2*z^2 + 1) * q^20 + (-2*z^3 - 2*z + 2) * q^22 + (2*z^3 + 2*z^2 - 2*z - 2) * q^23 + (5*z^3 - 5*z^2 + 5*z - 5) * q^25 + (-z^3 + z^2 - 4) * q^26 + (2*z^3 + 4*z - 4) * q^28 + (5*z^3 - z + 1) * q^29 + (2*z^3 + 4*z^2 + 2*z) * q^31 - 5 * q^32 + (3*z^2 - 2*z + 3) * q^34 - 10*z^2 * q^35 + (-5*z^2 - 5) * q^37 + (-2*z^2 - 2) * q^38 + (-3*z^3 - 3*z^2 + 3*z + 3) * q^40 + (z^2 - 3*z + 1) * q^41 + (6*z^3 - 6*z^2 - 2) * q^43 + (2*z^3 + 2*z) * q^44 + (2*z^3 + 4*z - 4) * q^46 + (-4*z^3 - 2*z + 2) * q^47 + 13 * q^49 + 5*z^3 * q^50 + (-4*z^3 + 5*z^2 - 5*z + 4) * q^52 + (2*z^3 - z + 1) * q^53 + (-2*z^2 - 4*z - 2) * q^55 + (12*z^3 - 6*z^2 + 12*z) * q^56 + (-z^3 + 6*z^2 - z) * q^58 - 4*z * q^59 + (-z^2 + z) * q^61 + (2*z^2 + 4*z + 2) * q^62 + (7*z^3 - 7*z^2 + 7*z - 7) * q^64 + (2*z^3 + 9*z - 9) * q^65 + (2*z^3 + 2*z^2 + 2*z) * q^67 + (3*z^3 - 3*z^2 - 1) * q^68 - 10*z * q^70 + (-2*z^3 + 2*z - 2) * q^71 + (5*z^2 - 5*z) * q^73 + (5*z^3 - 5*z^2 - 5) * q^74 + (-2*z^3 + 2*z^2 + 2) * q^76 + (-12*z^3 + 8*z^2 - 8*z + 12) * q^77 + (-z^3 - 2*z + 2) * q^80 + (-z^3 + z^2 - 2) * q^82 + (4*z^3 - 10*z^2 + 4*z) * q^83 + (-7*z^3 + 8*z^2 - 8*z + 7) * q^85 + (2*z^3 + 4*z^2 - 4*z - 2) * q^86 + (6*z^2 + 6) * q^88 + (-7*z^3 + 6*z^2 - 6*z + 7) * q^89 + (-18*z^2 + 14*z - 18) * q^91 + (-4*z^3 + 2*z^2 - 4*z) * q^92 + (-2*z^3 - 2*z^2 - 2*z) * q^94 + (6*z^3 - 4*z^2 + 4*z - 6) * q^95 + (-7*z^3 - 3*z + 3) * q^97 + (-13*z^3 + 13*z^2 - 13*z + 13) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + q^{2} + q^{4} - 5 q^{5} - 3 q^{8}+O(q^{10})$$ 4 * q + q^2 + q^4 - 5 * q^5 - 3 * q^8 $$4 q + q^{2} + q^{4} - 5 q^{5} - 3 q^{8} - 5 q^{10} + 6 q^{11} - 2 q^{13} + 10 q^{14} + q^{16} + 8 q^{17} - 4 q^{19} + 4 q^{22} - 10 q^{23} - 5 q^{25} - 18 q^{26} - 10 q^{28} + 8 q^{29} - 20 q^{32} + 7 q^{34} + 10 q^{35} - 15 q^{37} - 6 q^{38} + 15 q^{40} + 4 q^{43} + 4 q^{44} - 10 q^{46} + 2 q^{47} + 52 q^{49} + 5 q^{50} + 2 q^{52} + 5 q^{53} - 10 q^{55} + 30 q^{56} - 8 q^{58} - 4 q^{59} + 2 q^{61} + 10 q^{62} - 7 q^{64} - 25 q^{65} + 2 q^{67} + 2 q^{68} - 10 q^{70} - 8 q^{71} - 10 q^{73} - 10 q^{74} + 4 q^{76} + 20 q^{77} + 5 q^{80} - 10 q^{82} + 18 q^{83} + 5 q^{85} - 14 q^{86} + 18 q^{88} + 9 q^{89} - 40 q^{91} - 10 q^{92} - 2 q^{94} - 10 q^{95} + 2 q^{97} + 13 q^{98}+O(q^{100})$$ 4 * q + q^2 + q^4 - 5 * q^5 - 3 * q^8 - 5 * q^10 + 6 * q^11 - 2 * q^13 + 10 * q^14 + q^16 + 8 * q^17 - 4 * q^19 + 4 * q^22 - 10 * q^23 - 5 * q^25 - 18 * q^26 - 10 * q^28 + 8 * q^29 - 20 * q^32 + 7 * q^34 + 10 * q^35 - 15 * q^37 - 6 * q^38 + 15 * q^40 + 4 * q^43 + 4 * q^44 - 10 * q^46 + 2 * q^47 + 52 * q^49 + 5 * q^50 + 2 * q^52 + 5 * q^53 - 10 * q^55 + 30 * q^56 - 8 * q^58 - 4 * q^59 + 2 * q^61 + 10 * q^62 - 7 * q^64 - 25 * q^65 + 2 * q^67 + 2 * q^68 - 10 * q^70 - 8 * q^71 - 10 * q^73 - 10 * q^74 + 4 * q^76 + 20 * q^77 + 5 * q^80 - 10 * q^82 + 18 * q^83 + 5 * q^85 - 14 * q^86 + 18 * q^88 + 9 * q^89 - 40 * q^91 - 10 * q^92 - 2 * q^94 - 10 * q^95 + 2 * q^97 + 13 * 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
−0.309017 + 0.951057i 0 0.809017 + 0.587785i −1.80902 + 1.31433i 0 −4.47214 −2.42705 + 1.76336i 0 −0.690983 2.12663i
91.1 0.809017 + 0.587785i 0 −0.309017 0.951057i −0.690983 + 2.12663i 0 4.47214 0.927051 2.85317i 0 −1.80902 + 1.31433i
136.1 0.809017 0.587785i 0 −0.309017 + 0.951057i −0.690983 2.12663i 0 4.47214 0.927051 + 2.85317i 0 −1.80902 1.31433i
181.1 −0.309017 0.951057i 0 0.809017 0.587785i −1.80902 1.31433i 0 −4.47214 −2.42705 1.76336i 0 −0.690983 + 2.12663i
 $$n$$: 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.a 4
3.b odd 2 1 75.2.g.a 4
15.d odd 2 1 375.2.g.a 4
15.e even 4 2 375.2.i.a 8
25.d even 5 1 inner 225.2.h.a 4
25.d even 5 1 5625.2.a.a 2
25.e even 10 1 5625.2.a.h 2
75.h odd 10 1 375.2.g.a 4
75.h odd 10 1 1875.2.a.a 2
75.j odd 10 1 75.2.g.a 4
75.j odd 10 1 1875.2.a.d 2
75.l even 20 2 375.2.i.a 8
75.l even 20 2 1875.2.b.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.2.g.a 4 3.b odd 2 1
75.2.g.a 4 75.j odd 10 1
225.2.h.a 4 1.a even 1 1 trivial
225.2.h.a 4 25.d even 5 1 inner
375.2.g.a 4 15.d odd 2 1
375.2.g.a 4 75.h odd 10 1
375.2.i.a 8 15.e even 4 2
375.2.i.a 8 75.l even 20 2
1875.2.a.a 2 75.h odd 10 1
1875.2.a.d 2 75.j odd 10 1
1875.2.b.b 4 75.l even 20 2
5625.2.a.a 2 25.d even 5 1
5625.2.a.h 2 25.e even 10 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - T^{3} + T^{2} - T + 1$$
$3$ $$T^{4}$$
$5$ $$T^{4} + 5 T^{3} + 15 T^{2} + 25 T + 25$$
$7$ $$(T^{2} - 20)^{2}$$
$11$ $$T^{4} - 6 T^{3} + 16 T^{2} - 16 T + 16$$
$13$ $$T^{4} + 2 T^{3} + 24 T^{2} + 133 T + 361$$
$17$ $$T^{4} - 8 T^{3} + 34 T^{2} - 77 T + 121$$
$19$ $$T^{4} + 4 T^{3} + 16 T^{2} + 24 T + 16$$
$23$ $$T^{4} + 10 T^{3} + 60 T^{2} + \cdots + 400$$
$29$ $$T^{4} - 8 T^{3} + 34 T^{2} - 87 T + 841$$
$31$ $$T^{4} + 40 T^{2} + 200 T + 400$$
$37$ $$T^{4} + 15 T^{3} + 100 T^{2} + \cdots + 625$$
$41$ $$T^{4} + 10 T^{2} + 25 T + 25$$
$43$ $$(T^{2} - 2 T - 44)^{2}$$
$47$ $$T^{4} - 2 T^{3} + 24 T^{2} + 32 T + 16$$
$53$ $$T^{4} - 5 T^{3} + 10 T^{2} + 25$$
$59$ $$T^{4} + 4 T^{3} + 16 T^{2} + 64 T + 256$$
$61$ $$T^{4} - 2 T^{3} + 4 T^{2} - 3 T + 1$$
$67$ $$T^{4} - 2 T^{3} + 24 T^{2} + 32 T + 16$$
$71$ $$T^{4} + 8 T^{3} + 24 T^{2} - 8 T + 16$$
$73$ $$T^{4} + 10 T^{3} + 100 T^{2} + \cdots + 625$$
$79$ $$T^{4}$$
$83$ $$T^{4} - 18 T^{3} + 124 T^{2} + \cdots + 1936$$
$89$ $$T^{4} - 9 T^{3} + 46 T^{2} + \cdots + 1681$$
$97$ $$T^{4} - 2 T^{3} + 64 T^{2} + 247 T + 361$$