# Properties

 Label 900.2.k.c Level $900$ Weight $2$ Character orbit 900.k Analytic conductor $7.187$ Analytic rank $0$ Dimension $2$ CM discriminant -4 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [900,2,Mod(307,900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(900, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 0, 1]))

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

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

chi := DirichletCharacter("900.307");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$900 = 2^{2} \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 900.k (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: $$7.18653618192$$ 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: $$1$$ Twist minimal: no (minimal twist has level 20) Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

## $q$-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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - i - 1) q^{2} + 2 i q^{4} + ( - 2 i + 2) q^{8}+O(q^{10})$$ q + (-i - 1) * q^2 + 2*i * q^4 + (-2*i + 2) * q^8 $$q + ( - i - 1) q^{2} + 2 i q^{4} + ( - 2 i + 2) q^{8} + ( - i + 1) q^{13} - 4 q^{16} + (3 i + 3) q^{17} - 2 q^{26} - 4 i q^{29} + (4 i + 4) q^{32} - 6 i q^{34} + (7 i + 7) q^{37} + 8 q^{41} + 7 i q^{49} + (2 i + 2) q^{52} + ( - 9 i + 9) q^{53} + (4 i - 4) q^{58} + 12 q^{61} - 8 i q^{64} + (6 i - 6) q^{68} + ( - 11 i + 11) q^{73} - 14 i q^{74} + ( - 8 i - 8) q^{82} + 16 i q^{89} + ( - 13 i - 13) q^{97} + ( - 7 i + 7) q^{98} +O(q^{100})$$ q + (-i - 1) * q^2 + 2*i * q^4 + (-2*i + 2) * q^8 + (-i + 1) * q^13 - 4 * q^16 + (3*i + 3) * q^17 - 2 * q^26 - 4*i * q^29 + (4*i + 4) * q^32 - 6*i * q^34 + (7*i + 7) * q^37 + 8 * q^41 + 7*i * q^49 + (2*i + 2) * q^52 + (-9*i + 9) * q^53 + (4*i - 4) * q^58 + 12 * q^61 - 8*i * q^64 + (6*i - 6) * q^68 + (-11*i + 11) * q^73 - 14*i * q^74 + (-8*i - 8) * q^82 + 16*i * q^89 + (-13*i - 13) * q^97 + (-7*i + 7) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 4 q^{8}+O(q^{10})$$ 2 * q - 2 * q^2 + 4 * q^8 $$2 q - 2 q^{2} + 4 q^{8} + 2 q^{13} - 8 q^{16} + 6 q^{17} - 4 q^{26} + 8 q^{32} + 14 q^{37} + 16 q^{41} + 4 q^{52} + 18 q^{53} - 8 q^{58} + 24 q^{61} - 12 q^{68} + 22 q^{73} - 16 q^{82} - 26 q^{97} + 14 q^{98}+O(q^{100})$$ 2 * q - 2 * q^2 + 4 * q^8 + 2 * q^13 - 8 * q^16 + 6 * q^17 - 4 * q^26 + 8 * q^32 + 14 * q^37 + 16 * q^41 + 4 * q^52 + 18 * q^53 - 8 * q^58 + 24 * q^61 - 12 * q^68 + 22 * q^73 - 16 * q^82 - 26 * q^97 + 14 * q^98

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/900\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$451$$ $$577$$ $$\chi(n)$$ $$1$$ $$-1$$ $$i$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
307.1
 1.00000i − 1.00000i
−1.00000 1.00000i 0 2.00000i 0 0 0 2.00000 2.00000i 0 0
343.1 −1.00000 + 1.00000i 0 2.00000i 0 0 0 2.00000 + 2.00000i 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
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
5.c odd 4 1 inner
20.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.2.k.c 2
3.b odd 2 1 100.2.e.b 2
4.b odd 2 1 CM 900.2.k.c 2
5.b even 2 1 180.2.k.c 2
5.c odd 4 1 180.2.k.c 2
5.c odd 4 1 inner 900.2.k.c 2
12.b even 2 1 100.2.e.b 2
15.d odd 2 1 20.2.e.a 2
15.e even 4 1 20.2.e.a 2
15.e even 4 1 100.2.e.b 2
20.d odd 2 1 180.2.k.c 2
20.e even 4 1 180.2.k.c 2
20.e even 4 1 inner 900.2.k.c 2
24.f even 2 1 1600.2.n.h 2
24.h odd 2 1 1600.2.n.h 2
60.h even 2 1 20.2.e.a 2
60.l odd 4 1 20.2.e.a 2
60.l odd 4 1 100.2.e.b 2
105.g even 2 1 980.2.k.a 2
105.k odd 4 1 980.2.k.a 2
105.o odd 6 2 980.2.x.d 4
105.p even 6 2 980.2.x.c 4
105.w odd 12 2 980.2.x.c 4
105.x even 12 2 980.2.x.d 4
120.i odd 2 1 320.2.n.e 2
120.m even 2 1 320.2.n.e 2
120.q odd 4 1 320.2.n.e 2
120.q odd 4 1 1600.2.n.h 2
120.w even 4 1 320.2.n.e 2
120.w even 4 1 1600.2.n.h 2
240.t even 4 1 1280.2.o.g 2
240.t even 4 1 1280.2.o.j 2
240.z odd 4 1 1280.2.o.j 2
240.bb even 4 1 1280.2.o.j 2
240.bd odd 4 1 1280.2.o.g 2
240.bf even 4 1 1280.2.o.g 2
240.bm odd 4 1 1280.2.o.g 2
240.bm odd 4 1 1280.2.o.j 2
420.o odd 2 1 980.2.k.a 2
420.w even 4 1 980.2.k.a 2
420.ba even 6 2 980.2.x.d 4
420.be odd 6 2 980.2.x.c 4
420.bp odd 12 2 980.2.x.d 4
420.br even 12 2 980.2.x.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.2.e.a 2 15.d odd 2 1
20.2.e.a 2 15.e even 4 1
20.2.e.a 2 60.h even 2 1
20.2.e.a 2 60.l odd 4 1
100.2.e.b 2 3.b odd 2 1
100.2.e.b 2 12.b even 2 1
100.2.e.b 2 15.e even 4 1
100.2.e.b 2 60.l odd 4 1
180.2.k.c 2 5.b even 2 1
180.2.k.c 2 5.c odd 4 1
180.2.k.c 2 20.d odd 2 1
180.2.k.c 2 20.e even 4 1
320.2.n.e 2 120.i odd 2 1
320.2.n.e 2 120.m even 2 1
320.2.n.e 2 120.q odd 4 1
320.2.n.e 2 120.w even 4 1
900.2.k.c 2 1.a even 1 1 trivial
900.2.k.c 2 4.b odd 2 1 CM
900.2.k.c 2 5.c odd 4 1 inner
900.2.k.c 2 20.e even 4 1 inner
980.2.k.a 2 105.g even 2 1
980.2.k.a 2 105.k odd 4 1
980.2.k.a 2 420.o odd 2 1
980.2.k.a 2 420.w even 4 1
980.2.x.c 4 105.p even 6 2
980.2.x.c 4 105.w odd 12 2
980.2.x.c 4 420.be odd 6 2
980.2.x.c 4 420.br even 12 2
980.2.x.d 4 105.o odd 6 2
980.2.x.d 4 105.x even 12 2
980.2.x.d 4 420.ba even 6 2
980.2.x.d 4 420.bp odd 12 2
1280.2.o.g 2 240.t even 4 1
1280.2.o.g 2 240.bd odd 4 1
1280.2.o.g 2 240.bf even 4 1
1280.2.o.g 2 240.bm odd 4 1
1280.2.o.j 2 240.t even 4 1
1280.2.o.j 2 240.z odd 4 1
1280.2.o.j 2 240.bb even 4 1
1280.2.o.j 2 240.bm odd 4 1
1600.2.n.h 2 24.f even 2 1
1600.2.n.h 2 24.h odd 2 1
1600.2.n.h 2 120.q odd 4 1
1600.2.n.h 2 120.w even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(900, [\chi])$$:

 $$T_{7}$$ T7 $$T_{11}$$ T11 $$T_{13}^{2} - 2T_{13} + 2$$ T13^2 - 2*T13 + 2 $$T_{17}^{2} - 6T_{17} + 18$$ T17^2 - 6*T17 + 18 $$T_{19}$$ T19 $$T_{23}$$ T23

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 2T + 2$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2} - 2T + 2$$
$17$ $$T^{2} - 6T + 18$$
$19$ $$T^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2} + 16$$
$31$ $$T^{2}$$
$37$ $$T^{2} - 14T + 98$$
$41$ $$(T - 8)^{2}$$
$43$ $$T^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2} - 18T + 162$$
$59$ $$T^{2}$$
$61$ $$(T - 12)^{2}$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$T^{2} - 22T + 242$$
$79$ $$T^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2} + 256$$
$97$ $$T^{2} + 26T + 338$$