# Properties

 Label 225.3.g.a Level $225$ Weight $3$ Character orbit 225.g Analytic conductor $6.131$ 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] = [225,3,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 15) 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{2} + \beta_1 - 1) q^{2} + ( - 2 \beta_{3} + \beta_{2} - 2 \beta_1) q^{4} + ( - \beta_{2} - 2 \beta_1 - 1) q^{7} + (\beta_{3} - 3 \beta_{2} + 3) q^{8}+O(q^{10})$$ q + (-b2 + b1 - 1) * q^2 + (-2*b3 + b2 - 2*b1) * q^4 + (-b2 - 2*b1 - 1) * q^7 + (b3 - 3*b2 + 3) * q^8 $$q + ( - \beta_{2} + \beta_1 - 1) q^{2} + ( - 2 \beta_{3} + \beta_{2} - 2 \beta_1) q^{4} + ( - \beta_{2} - 2 \beta_1 - 1) q^{7} + (\beta_{3} - 3 \beta_{2} + 3) q^{8} + (3 \beta_{3} - 3 \beta_1 - 4) q^{11} + (2 \beta_{3} - 8 \beta_{2} + 8) q^{13} + (\beta_{3} - 4 \beta_{2} + \beta_1) q^{14} + (4 \beta_{3} - 4 \beta_1 - 5) q^{16} + ( - 10 \beta_{2} - 6 \beta_1 - 10) q^{17} + (6 \beta_{3} - 6 \beta_{2} + 6 \beta_1) q^{19} + ( - 5 \beta_{2} + 2 \beta_1 - 5) q^{22} + (2 \beta_{3} - 14 \beta_{2} + 14) q^{23} + ( - 10 \beta_{3} + 10 \beta_1 - 22) q^{26} + (2 \beta_{3} + 11 \beta_{2} - 11) q^{28} + ( - 7 \beta_{3} - 18 \beta_{2} - 7 \beta_1) q^{29} + ( - 6 \beta_{3} + 6 \beta_1 - 4) q^{31} + ( - 19 \beta_{2} + 7 \beta_1 - 19) q^{32} + ( - 4 \beta_{3} + 2 \beta_{2} - 4 \beta_1) q^{34} + ( - 16 \beta_{2} + 18 \beta_1 - 16) q^{37} + ( - 18 \beta_{3} + 24 \beta_{2} - 24) q^{38} + (6 \beta_{3} - 6 \beta_1 + 14) q^{41} + ( - 20 \beta_{3} - 2 \beta_{2} + 2) q^{43} + (5 \beta_{3} + 32 \beta_{2} + 5 \beta_1) q^{44} + ( - 16 \beta_{3} + 16 \beta_1 - 34) q^{46} + (32 \beta_{2} + 10 \beta_1 + 32) q^{47} + (4 \beta_{3} - 35 \beta_{2} + 4 \beta_1) q^{49} + (20 \beta_{2} - 34 \beta_1 + 20) q^{52} + (12 \beta_{3} - 14 \beta_{2} + 14) q^{53} + (5 \beta_{3} - 5 \beta_1) q^{56} + ( - 4 \beta_{3} - 3 \beta_{2} + 3) q^{58} + (31 \beta_{3} - 36 \beta_{2} + 31 \beta_1) q^{59} + (18 \beta_{3} - 18 \beta_1 + 50) q^{61} + (22 \beta_{2} - 16 \beta_1 + 22) q^{62} + ( - 10 \beta_{3} + 79 \beta_{2} - 10 \beta_1) q^{64} + (50 \beta_{2} - 4 \beta_1 + 50) q^{67} + (34 \beta_{3} + 26 \beta_{2} - 26) q^{68} + 68 q^{71} + (48 \beta_{3} + 19 \beta_{2} - 19) q^{73} + ( - 34 \beta_{3} + 86 \beta_{2} - 34 \beta_1) q^{74} + (18 \beta_{3} - 18 \beta_1 + 78) q^{76} + (22 \beta_{2} + 14 \beta_1 + 22) q^{77} + ( - 10 \beta_{3} - 10 \beta_1) q^{79} + ( - 32 \beta_{2} + 26 \beta_1 - 32) q^{82} + ( - 14 \beta_{3} + 4 \beta_{2} - 4) q^{83} + (18 \beta_{3} - 18 \beta_1 + 56) q^{86} + (14 \beta_{3} + 3 \beta_{2} - 3) q^{88} + ( - 36 \beta_{3} + 6 \beta_{2} - 36 \beta_1) q^{89} + (14 \beta_{3} - 14 \beta_1 - 4) q^{91} + (26 \beta_{2} - 58 \beta_1 + 26) q^{92} + (22 \beta_{3} - 34 \beta_{2} + 22 \beta_1) q^{94} + (5 \beta_{2} + 16 \beta_1 + 5) q^{97} + ( - 43 \beta_{3} + 47 \beta_{2} - 47) q^{98}+O(q^{100})$$ q + (-b2 + b1 - 1) * q^2 + (-2*b3 + b2 - 2*b1) * q^4 + (-b2 - 2*b1 - 1) * q^7 + (b3 - 3*b2 + 3) * q^8 + (3*b3 - 3*b1 - 4) * q^11 + (2*b3 - 8*b2 + 8) * q^13 + (b3 - 4*b2 + b1) * q^14 + (4*b3 - 4*b1 - 5) * q^16 + (-10*b2 - 6*b1 - 10) * q^17 + (6*b3 - 6*b2 + 6*b1) * q^19 + (-5*b2 + 2*b1 - 5) * q^22 + (2*b3 - 14*b2 + 14) * q^23 + (-10*b3 + 10*b1 - 22) * q^26 + (2*b3 + 11*b2 - 11) * q^28 + (-7*b3 - 18*b2 - 7*b1) * q^29 + (-6*b3 + 6*b1 - 4) * q^31 + (-19*b2 + 7*b1 - 19) * q^32 + (-4*b3 + 2*b2 - 4*b1) * q^34 + (-16*b2 + 18*b1 - 16) * q^37 + (-18*b3 + 24*b2 - 24) * q^38 + (6*b3 - 6*b1 + 14) * q^41 + (-20*b3 - 2*b2 + 2) * q^43 + (5*b3 + 32*b2 + 5*b1) * q^44 + (-16*b3 + 16*b1 - 34) * q^46 + (32*b2 + 10*b1 + 32) * q^47 + (4*b3 - 35*b2 + 4*b1) * q^49 + (20*b2 - 34*b1 + 20) * q^52 + (12*b3 - 14*b2 + 14) * q^53 + (5*b3 - 5*b1) * q^56 + (-4*b3 - 3*b2 + 3) * q^58 + (31*b3 - 36*b2 + 31*b1) * q^59 + (18*b3 - 18*b1 + 50) * q^61 + (22*b2 - 16*b1 + 22) * q^62 + (-10*b3 + 79*b2 - 10*b1) * q^64 + (50*b2 - 4*b1 + 50) * q^67 + (34*b3 + 26*b2 - 26) * q^68 + 68 * q^71 + (48*b3 + 19*b2 - 19) * q^73 + (-34*b3 + 86*b2 - 34*b1) * q^74 + (18*b3 - 18*b1 + 78) * q^76 + (22*b2 + 14*b1 + 22) * q^77 + (-10*b3 - 10*b1) * q^79 + (-32*b2 + 26*b1 - 32) * q^82 + (-14*b3 + 4*b2 - 4) * q^83 + (18*b3 - 18*b1 + 56) * q^86 + (14*b3 + 3*b2 - 3) * q^88 + (-36*b3 + 6*b2 - 36*b1) * q^89 + (14*b3 - 14*b1 - 4) * q^91 + (26*b2 - 58*b1 + 26) * q^92 + (22*b3 - 34*b2 + 22*b1) * q^94 + (5*b2 + 16*b1 + 5) * q^97 + (-43*b3 + 47*b2 - 47) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{2} - 4 q^{7} + 12 q^{8}+O(q^{10})$$ 4 * q - 4 * q^2 - 4 * q^7 + 12 * q^8 $$4 q - 4 q^{2} - 4 q^{7} + 12 q^{8} - 16 q^{11} + 32 q^{13} - 20 q^{16} - 40 q^{17} - 20 q^{22} + 56 q^{23} - 88 q^{26} - 44 q^{28} - 16 q^{31} - 76 q^{32} - 64 q^{37} - 96 q^{38} + 56 q^{41} + 8 q^{43} - 136 q^{46} + 128 q^{47} + 80 q^{52} + 56 q^{53} + 12 q^{58} + 200 q^{61} + 88 q^{62} + 200 q^{67} - 104 q^{68} + 272 q^{71} - 76 q^{73} + 312 q^{76} + 88 q^{77} - 128 q^{82} - 16 q^{83} + 224 q^{86} - 12 q^{88} - 16 q^{91} + 104 q^{92} + 20 q^{97} - 188 q^{98}+O(q^{100})$$ 4 * q - 4 * q^2 - 4 * q^7 + 12 * q^8 - 16 * q^11 + 32 * q^13 - 20 * q^16 - 40 * q^17 - 20 * q^22 + 56 * q^23 - 88 * q^26 - 44 * q^28 - 16 * q^31 - 76 * q^32 - 64 * q^37 - 96 * q^38 + 56 * q^41 + 8 * q^43 - 136 * q^46 + 128 * q^47 + 80 * q^52 + 56 * q^53 + 12 * q^58 + 200 * q^61 + 88 * q^62 + 200 * q^67 - 104 * q^68 + 272 * q^71 - 76 * q^73 + 312 * q^76 + 88 * q^77 - 128 * q^82 - 16 * q^83 + 224 * q^86 - 12 * q^88 - 16 * q^91 + 104 * q^92 + 20 * q^97 - 188 * q^98

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
−2.22474 2.22474i 0 5.89898i 0 0 1.44949 + 1.44949i 4.22474 4.22474i 0 0
82.2 0.224745 + 0.224745i 0 3.89898i 0 0 −3.44949 3.44949i 1.77526 1.77526i 0 0
118.1 −2.22474 + 2.22474i 0 5.89898i 0 0 1.44949 1.44949i 4.22474 + 4.22474i 0 0
118.2 0.224745 0.224745i 0 3.89898i 0 0 −3.44949 + 3.44949i 1.77526 + 1.77526i 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.c odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.3.g.a 4
3.b odd 2 1 75.3.f.c 4
5.b even 2 1 45.3.g.b 4
5.c odd 4 1 45.3.g.b 4
5.c odd 4 1 inner 225.3.g.a 4
12.b even 2 1 1200.3.bg.k 4
15.d odd 2 1 15.3.f.a 4
15.e even 4 1 15.3.f.a 4
15.e even 4 1 75.3.f.c 4
20.d odd 2 1 720.3.bh.k 4
20.e even 4 1 720.3.bh.k 4
45.h odd 6 2 405.3.l.h 8
45.j even 6 2 405.3.l.f 8
45.k odd 12 2 405.3.l.f 8
45.l even 12 2 405.3.l.h 8
60.h even 2 1 240.3.bg.a 4
60.l odd 4 1 240.3.bg.a 4
60.l odd 4 1 1200.3.bg.k 4
120.i odd 2 1 960.3.bg.i 4
120.m even 2 1 960.3.bg.h 4
120.q odd 4 1 960.3.bg.h 4
120.w even 4 1 960.3.bg.i 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.3.f.a 4 15.d odd 2 1
15.3.f.a 4 15.e even 4 1
45.3.g.b 4 5.b even 2 1
45.3.g.b 4 5.c odd 4 1
75.3.f.c 4 3.b odd 2 1
75.3.f.c 4 15.e even 4 1
225.3.g.a 4 1.a even 1 1 trivial
225.3.g.a 4 5.c odd 4 1 inner
240.3.bg.a 4 60.h even 2 1
240.3.bg.a 4 60.l odd 4 1
405.3.l.f 8 45.j even 6 2
405.3.l.f 8 45.k odd 12 2
405.3.l.h 8 45.h odd 6 2
405.3.l.h 8 45.l even 12 2
720.3.bh.k 4 20.d odd 2 1
720.3.bh.k 4 20.e even 4 1
960.3.bg.h 4 120.m even 2 1
960.3.bg.h 4 120.q odd 4 1
960.3.bg.i 4 120.i odd 2 1
960.3.bg.i 4 120.w even 4 1
1200.3.bg.k 4 12.b even 2 1
1200.3.bg.k 4 60.l odd 4 1

## Hecke kernels

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

 $$T_{2}^{4} + 4T_{2}^{3} + 8T_{2}^{2} - 4T_{2} + 1$$ T2^4 + 4*T2^3 + 8*T2^2 - 4*T2 + 1 $$T_{7}^{4} + 4T_{7}^{3} + 8T_{7}^{2} - 40T_{7} + 100$$ T7^4 + 4*T7^3 + 8*T7^2 - 40*T7 + 100

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 4 T^{3} + 8 T^{2} - 4 T + 1$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 4 T^{3} + 8 T^{2} - 40 T + 100$$
$11$ $$(T^{2} + 8 T - 38)^{2}$$
$13$ $$T^{4} - 32 T^{3} + 512 T^{2} + \cdots + 13456$$
$17$ $$T^{4} + 40 T^{3} + 800 T^{2} + \cdots + 8464$$
$19$ $$T^{4} + 504 T^{2} + 32400$$
$23$ $$T^{4} - 56 T^{3} + 1568 T^{2} + \cdots + 144400$$
$29$ $$T^{4} + 1236T^{2} + 900$$
$31$ $$(T^{2} + 8 T - 200)^{2}$$
$37$ $$T^{4} + 64 T^{3} + 2048 T^{2} + \cdots + 211600$$
$41$ $$(T^{2} - 28 T - 20)^{2}$$
$43$ $$T^{4} - 8 T^{3} + 32 T^{2} + \cdots + 1420864$$
$47$ $$T^{4} - 128 T^{3} + 8192 T^{2} + \cdots + 3055504$$
$53$ $$T^{4} - 56 T^{3} + 1568 T^{2} + \cdots + 1600$$
$59$ $$T^{4} + 14124 T^{2} + \cdots + 19980900$$
$61$ $$(T^{2} - 100 T + 556)^{2}$$
$67$ $$T^{4} - 200 T^{3} + \cdots + 24522304$$
$71$ $$(T - 68)^{4}$$
$73$ $$T^{4} + 76 T^{3} + 2888 T^{2} + \cdots + 38316100$$
$79$ $$(T^{2} + 600)^{2}$$
$83$ $$T^{4} + 16 T^{3} + 128 T^{2} + \cdots + 309136$$
$89$ $$T^{4} + 15624 T^{2} + \cdots + 59907600$$
$97$ $$T^{4} - 20 T^{3} + 200 T^{2} + \cdots + 515524$$