# Properties

 Label 225.3.g.e Level $225$ Weight $3$ Character orbit 225.g Analytic conductor $6.131$ 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] = [225,3,Mod(82,225)]

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([0, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("225.82");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$225 = 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 225.g (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: $$6.13080594811$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(i, \sqrt{6})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 9$$ x^4 + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 25) 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 basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} - \beta_{2} q^{4} + 4 \beta_1 q^{7} - 5 \beta_{3} q^{8}+O(q^{10})$$ q + b1 * q^2 - b2 * q^4 + 4*b1 * q^7 - 5*b3 * q^8 $$q + \beta_1 q^{2} - \beta_{2} q^{4} + 4 \beta_1 q^{7} - 5 \beta_{3} q^{8} + 3 q^{11} - 6 \beta_{3} q^{13} + 12 \beta_{2} q^{14} + 11 q^{16} + 11 \beta_1 q^{17} + 5 \beta_{2} q^{19} + 3 \beta_1 q^{22} - 14 \beta_{3} q^{23} + 18 q^{26} - 4 \beta_{3} q^{28} + 30 \beta_{2} q^{29} - 38 q^{31} - 9 \beta_1 q^{32} + 33 \beta_{2} q^{34} - 16 \beta_1 q^{37} + 5 \beta_{3} q^{38} - 57 q^{41} + 4 \beta_{3} q^{43} - 3 \beta_{2} q^{44} + 42 q^{46} + 6 \beta_1 q^{47} - \beta_{2} q^{49} - 6 \beta_1 q^{52} + 26 \beta_{3} q^{53} + 60 q^{56} + 30 \beta_{3} q^{58} - 90 \beta_{2} q^{59} - 28 q^{61} - 38 \beta_1 q^{62} - 71 \beta_{2} q^{64} + 39 \beta_1 q^{67} - 11 \beta_{3} q^{68} - 42 q^{71} - 11 \beta_{3} q^{73} - 48 \beta_{2} q^{74} + 5 q^{76} + 12 \beta_1 q^{77} - 80 \beta_{2} q^{79} - 57 \beta_1 q^{82} + 91 \beta_{3} q^{83} - 12 q^{86} - 15 \beta_{3} q^{88} + 15 \beta_{2} q^{89} + 72 q^{91} - 14 \beta_1 q^{92} + 18 \beta_{2} q^{94} + 44 \beta_1 q^{97} - \beta_{3} q^{98}+O(q^{100})$$ q + b1 * q^2 - b2 * q^4 + 4*b1 * q^7 - 5*b3 * q^8 + 3 * q^11 - 6*b3 * q^13 + 12*b2 * q^14 + 11 * q^16 + 11*b1 * q^17 + 5*b2 * q^19 + 3*b1 * q^22 - 14*b3 * q^23 + 18 * q^26 - 4*b3 * q^28 + 30*b2 * q^29 - 38 * q^31 - 9*b1 * q^32 + 33*b2 * q^34 - 16*b1 * q^37 + 5*b3 * q^38 - 57 * q^41 + 4*b3 * q^43 - 3*b2 * q^44 + 42 * q^46 + 6*b1 * q^47 - b2 * q^49 - 6*b1 * q^52 + 26*b3 * q^53 + 60 * q^56 + 30*b3 * q^58 - 90*b2 * q^59 - 28 * q^61 - 38*b1 * q^62 - 71*b2 * q^64 + 39*b1 * q^67 - 11*b3 * q^68 - 42 * q^71 - 11*b3 * q^73 - 48*b2 * q^74 + 5 * q^76 + 12*b1 * q^77 - 80*b2 * q^79 - 57*b1 * q^82 + 91*b3 * q^83 - 12 * q^86 - 15*b3 * q^88 + 15*b2 * q^89 + 72 * q^91 - 14*b1 * q^92 + 18*b2 * q^94 + 44*b1 * q^97 - b3 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q + 12 q^{11} + 44 q^{16} + 72 q^{26} - 152 q^{31} - 228 q^{41} + 168 q^{46} + 240 q^{56} - 112 q^{61} - 168 q^{71} + 20 q^{76} - 48 q^{86} + 288 q^{91}+O(q^{100})$$ 4 * q + 12 * q^11 + 44 * q^16 + 72 * q^26 - 152 * q^31 - 228 * q^41 + 168 * q^46 + 240 * q^56 - 112 * q^61 - 168 * q^71 + 20 * q^76 - 48 * q^86 + 288 * q^91

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 9$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 3$$ (v^2) / 3 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 3$$ (v^3) / 3
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$3\beta_{2}$$ 3*b2 $$\nu^{3}$$ $$=$$ $$3\beta_{3}$$ 3*b3

## Character values

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

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$1$$ $$\beta_{2}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
82.1
 −1.22474 − 1.22474i 1.22474 + 1.22474i −1.22474 + 1.22474i 1.22474 − 1.22474i
−1.22474 1.22474i 0 1.00000i 0 0 −4.89898 4.89898i −6.12372 + 6.12372i 0 0
82.2 1.22474 + 1.22474i 0 1.00000i 0 0 4.89898 + 4.89898i 6.12372 6.12372i 0 0
118.1 −1.22474 + 1.22474i 0 1.00000i 0 0 −4.89898 + 4.89898i −6.12372 6.12372i 0 0
118.2 1.22474 1.22474i 0 1.00000i 0 0 4.89898 4.89898i 6.12372 + 6.12372i 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
5.b even 2 1 inner
5.c odd 4 2 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.3.g.e 4
3.b odd 2 1 25.3.c.a 4
5.b even 2 1 inner 225.3.g.e 4
5.c odd 4 2 inner 225.3.g.e 4
12.b even 2 1 400.3.p.j 4
15.d odd 2 1 25.3.c.a 4
15.e even 4 2 25.3.c.a 4
60.h even 2 1 400.3.p.j 4
60.l odd 4 2 400.3.p.j 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.3.c.a 4 3.b odd 2 1
25.3.c.a 4 15.d odd 2 1
25.3.c.a 4 15.e even 4 2
225.3.g.e 4 1.a even 1 1 trivial
225.3.g.e 4 5.b even 2 1 inner
225.3.g.e 4 5.c odd 4 2 inner
400.3.p.j 4 12.b even 2 1
400.3.p.j 4 60.h even 2 1
400.3.p.j 4 60.l odd 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(225, [\chi])$$:

 $$T_{2}^{4} + 9$$ T2^4 + 9 $$T_{7}^{4} + 2304$$ T7^4 + 2304

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 9$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 2304$$
$11$ $$(T - 3)^{4}$$
$13$ $$T^{4} + 11664$$
$17$ $$T^{4} + 131769$$
$19$ $$(T^{2} + 25)^{2}$$
$23$ $$T^{4} + 345744$$
$29$ $$(T^{2} + 900)^{2}$$
$31$ $$(T + 38)^{4}$$
$37$ $$T^{4} + 589824$$
$41$ $$(T + 57)^{4}$$
$43$ $$T^{4} + 2304$$
$47$ $$T^{4} + 11664$$
$53$ $$T^{4} + 4112784$$
$59$ $$(T^{2} + 8100)^{2}$$
$61$ $$(T + 28)^{4}$$
$67$ $$T^{4} + 20820969$$
$71$ $$(T + 42)^{4}$$
$73$ $$T^{4} + 131769$$
$79$ $$(T^{2} + 6400)^{2}$$
$83$ $$T^{4} + 617174649$$
$89$ $$(T^{2} + 225)^{2}$$
$97$ $$T^{4} + 33732864$$