# Properties

 Label 225.4.b.b Level $225$ Weight $4$ Character orbit 225.b Analytic conductor $13.275$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,4,Mod(199,225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(225, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("225.199");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$225 = 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 225.b (of order $$2$$, degree $$1$$, 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: $$13.2754297513$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$5$$ Twist minimal: no (minimal twist has level 45) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 $$\beta = 5i$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} - 17 q^{4} - 6 \beta q^{7} - 9 \beta q^{8} +O(q^{10})$$ q + b * q^2 - 17 * q^4 - 6*b * q^7 - 9*b * q^8 $$q + \beta q^{2} - 17 q^{4} - 6 \beta q^{7} - 9 \beta q^{8} + 50 q^{11} + 4 \beta q^{13} + 150 q^{14} + 89 q^{16} - 2 \beta q^{17} + 44 q^{19} + 50 \beta q^{22} - 24 \beta q^{23} - 100 q^{26} + 102 \beta q^{28} + 50 q^{29} + 108 q^{31} + 17 \beta q^{32} + 50 q^{34} - 8 \beta q^{37} + 44 \beta q^{38} + 400 q^{41} - 56 \beta q^{43} - 850 q^{44} + 600 q^{46} - 56 \beta q^{47} - 557 q^{49} - 68 \beta q^{52} + 122 \beta q^{53} - 1350 q^{56} + 50 \beta q^{58} - 50 q^{59} - 518 q^{61} + 108 \beta q^{62} + 287 q^{64} - 36 \beta q^{67} + 34 \beta q^{68} + 700 q^{71} + 82 \beta q^{73} + 200 q^{74} - 748 q^{76} - 300 \beta q^{77} + 516 q^{79} + 400 \beta q^{82} - 132 \beta q^{83} + 1400 q^{86} - 450 \beta q^{88} + 1500 q^{89} + 600 q^{91} + 408 \beta q^{92} + 1400 q^{94} - 326 \beta q^{97} - 557 \beta q^{98} +O(q^{100})$$ q + b * q^2 - 17 * q^4 - 6*b * q^7 - 9*b * q^8 + 50 * q^11 + 4*b * q^13 + 150 * q^14 + 89 * q^16 - 2*b * q^17 + 44 * q^19 + 50*b * q^22 - 24*b * q^23 - 100 * q^26 + 102*b * q^28 + 50 * q^29 + 108 * q^31 + 17*b * q^32 + 50 * q^34 - 8*b * q^37 + 44*b * q^38 + 400 * q^41 - 56*b * q^43 - 850 * q^44 + 600 * q^46 - 56*b * q^47 - 557 * q^49 - 68*b * q^52 + 122*b * q^53 - 1350 * q^56 + 50*b * q^58 - 50 * q^59 - 518 * q^61 + 108*b * q^62 + 287 * q^64 - 36*b * q^67 + 34*b * q^68 + 700 * q^71 + 82*b * q^73 + 200 * q^74 - 748 * q^76 - 300*b * q^77 + 516 * q^79 + 400*b * q^82 - 132*b * q^83 + 1400 * q^86 - 450*b * q^88 + 1500 * q^89 + 600 * q^91 + 408*b * q^92 + 1400 * q^94 - 326*b * q^97 - 557*b * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 34 q^{4}+O(q^{10})$$ 2 * q - 34 * q^4 $$2 q - 34 q^{4} + 100 q^{11} + 300 q^{14} + 178 q^{16} + 88 q^{19} - 200 q^{26} + 100 q^{29} + 216 q^{31} + 100 q^{34} + 800 q^{41} - 1700 q^{44} + 1200 q^{46} - 1114 q^{49} - 2700 q^{56} - 100 q^{59} - 1036 q^{61} + 574 q^{64} + 1400 q^{71} + 400 q^{74} - 1496 q^{76} + 1032 q^{79} + 2800 q^{86} + 3000 q^{89} + 1200 q^{91} + 2800 q^{94}+O(q^{100})$$ 2 * q - 34 * q^4 + 100 * q^11 + 300 * q^14 + 178 * q^16 + 88 * q^19 - 200 * q^26 + 100 * q^29 + 216 * q^31 + 100 * q^34 + 800 * q^41 - 1700 * q^44 + 1200 * q^46 - 1114 * q^49 - 2700 * q^56 - 100 * q^59 - 1036 * q^61 + 574 * q^64 + 1400 * q^71 + 400 * q^74 - 1496 * q^76 + 1032 * q^79 + 2800 * q^86 + 3000 * q^89 + 1200 * q^91 + 2800 * q^94

## Character values

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

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
199.1
 − 1.00000i 1.00000i
5.00000i 0 −17.0000 0 0 30.0000i 45.0000i 0 0
199.2 5.00000i 0 −17.0000 0 0 30.0000i 45.0000i 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.4.b.b 2
3.b odd 2 1 225.4.b.a 2
5.b even 2 1 inner 225.4.b.b 2
5.c odd 4 1 45.4.a.e yes 1
5.c odd 4 1 225.4.a.a 1
15.d odd 2 1 225.4.b.a 2
15.e even 4 1 45.4.a.a 1
15.e even 4 1 225.4.a.h 1
20.e even 4 1 720.4.a.o 1
35.f even 4 1 2205.4.a.t 1
45.k odd 12 2 405.4.e.b 2
45.l even 12 2 405.4.e.n 2
60.l odd 4 1 720.4.a.bc 1
105.k odd 4 1 2205.4.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.4.a.a 1 15.e even 4 1
45.4.a.e yes 1 5.c odd 4 1
225.4.a.a 1 5.c odd 4 1
225.4.a.h 1 15.e even 4 1
225.4.b.a 2 3.b odd 2 1
225.4.b.a 2 15.d odd 2 1
225.4.b.b 2 1.a even 1 1 trivial
225.4.b.b 2 5.b even 2 1 inner
405.4.e.b 2 45.k odd 12 2
405.4.e.n 2 45.l even 12 2
720.4.a.o 1 20.e even 4 1
720.4.a.bc 1 60.l odd 4 1
2205.4.a.a 1 105.k odd 4 1
2205.4.a.t 1 35.f even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(225, [\chi])$$:

 $$T_{2}^{2} + 25$$ T2^2 + 25 $$T_{7}^{2} + 900$$ T7^2 + 900 $$T_{11} - 50$$ T11 - 50

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 25$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 900$$
$11$ $$(T - 50)^{2}$$
$13$ $$T^{2} + 400$$
$17$ $$T^{2} + 100$$
$19$ $$(T - 44)^{2}$$
$23$ $$T^{2} + 14400$$
$29$ $$(T - 50)^{2}$$
$31$ $$(T - 108)^{2}$$
$37$ $$T^{2} + 1600$$
$41$ $$(T - 400)^{2}$$
$43$ $$T^{2} + 78400$$
$47$ $$T^{2} + 78400$$
$53$ $$T^{2} + 372100$$
$59$ $$(T + 50)^{2}$$
$61$ $$(T + 518)^{2}$$
$67$ $$T^{2} + 32400$$
$71$ $$(T - 700)^{2}$$
$73$ $$T^{2} + 168100$$
$79$ $$(T - 516)^{2}$$
$83$ $$T^{2} + 435600$$
$89$ $$(T - 1500)^{2}$$
$97$ $$T^{2} + 2656900$$