# Properties

 Label 900.3.c.h Level $900$ Weight $3$ Character orbit 900.c Analytic conductor $24.523$ Analytic rank $0$ Dimension $2$ CM discriminant -20 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [900,3,Mod(451,900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(900, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("900.451");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$900 = 2^{2} \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 900.c (of order $$2$$, degree $$1$$, 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: $$24.5232237924$$ 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: $$2$$ Twist minimal: no (minimal twist has level 20) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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 $$\beta = 2i$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta q^{2} - 4 q^{4} + 2 \beta q^{7} + 4 \beta q^{8} +O(q^{10})$$ q - b * q^2 - 4 * q^4 + 2*b * q^7 + 4*b * q^8 $$q - \beta q^{2} - 4 q^{4} + 2 \beta q^{7} + 4 \beta q^{8} + 8 q^{14} + 16 q^{16} - 22 \beta q^{23} - 8 \beta q^{28} - 22 q^{29} - 16 \beta q^{32} - 62 q^{41} - 38 \beta q^{43} - 88 q^{46} - 2 \beta q^{47} + 33 q^{49} - 32 q^{56} + 22 \beta q^{58} - 58 q^{61} - 64 q^{64} - 58 \beta q^{67} + 62 \beta q^{82} + 38 \beta q^{83} - 152 q^{86} - 142 q^{89} + 88 \beta q^{92} - 8 q^{94} - 33 \beta q^{98} +O(q^{100})$$ q - b * q^2 - 4 * q^4 + 2*b * q^7 + 4*b * q^8 + 8 * q^14 + 16 * q^16 - 22*b * q^23 - 8*b * q^28 - 22 * q^29 - 16*b * q^32 - 62 * q^41 - 38*b * q^43 - 88 * q^46 - 2*b * q^47 + 33 * q^49 - 32 * q^56 + 22*b * q^58 - 58 * q^61 - 64 * q^64 - 58*b * q^67 + 62*b * q^82 + 38*b * q^83 - 152 * q^86 - 142 * q^89 + 88*b * q^92 - 8 * q^94 - 33*b * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 8 q^{4}+O(q^{10})$$ 2 * q - 8 * q^4 $$2 q - 8 q^{4} + 16 q^{14} + 32 q^{16} - 44 q^{29} - 124 q^{41} - 176 q^{46} + 66 q^{49} - 64 q^{56} - 116 q^{61} - 128 q^{64} - 304 q^{86} - 284 q^{89} - 16 q^{94}+O(q^{100})$$ 2 * q - 8 * q^4 + 16 * q^14 + 32 * q^16 - 44 * q^29 - 124 * q^41 - 176 * q^46 + 66 * q^49 - 64 * q^56 - 116 * q^61 - 128 * q^64 - 304 * q^86 - 284 * q^89 - 16 * q^94

## Character values

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

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

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
451.1
 1.00000i − 1.00000i
2.00000i 0 −4.00000 0 0 4.00000i 8.00000i 0 0
451.2 2.00000i 0 −4.00000 0 0 4.00000i 8.00000i 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
20.d odd 2 1 CM by $$\Q(\sqrt{-5})$$
4.b odd 2 1 inner
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 900.3.c.h 2
3.b odd 2 1 100.3.b.c 2
4.b odd 2 1 inner 900.3.c.h 2
5.b even 2 1 inner 900.3.c.h 2
5.c odd 4 1 180.3.f.a 1
5.c odd 4 1 180.3.f.b 1
12.b even 2 1 100.3.b.c 2
15.d odd 2 1 100.3.b.c 2
15.e even 4 1 20.3.d.a 1
15.e even 4 1 20.3.d.b yes 1
20.d odd 2 1 CM 900.3.c.h 2
20.e even 4 1 180.3.f.a 1
20.e even 4 1 180.3.f.b 1
24.f even 2 1 1600.3.b.f 2
24.h odd 2 1 1600.3.b.f 2
60.h even 2 1 100.3.b.c 2
60.l odd 4 1 20.3.d.a 1
60.l odd 4 1 20.3.d.b yes 1
120.i odd 2 1 1600.3.b.f 2
120.m even 2 1 1600.3.b.f 2
120.q odd 4 1 320.3.h.a 1
120.q odd 4 1 320.3.h.b 1
120.w even 4 1 320.3.h.a 1
120.w even 4 1 320.3.h.b 1
240.z odd 4 1 1280.3.e.b 2
240.z odd 4 1 1280.3.e.c 2
240.bb even 4 1 1280.3.e.b 2
240.bb even 4 1 1280.3.e.c 2
240.bd odd 4 1 1280.3.e.b 2
240.bd odd 4 1 1280.3.e.c 2
240.bf even 4 1 1280.3.e.b 2
240.bf even 4 1 1280.3.e.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.3.d.a 1 15.e even 4 1
20.3.d.a 1 60.l odd 4 1
20.3.d.b yes 1 15.e even 4 1
20.3.d.b yes 1 60.l odd 4 1
100.3.b.c 2 3.b odd 2 1
100.3.b.c 2 12.b even 2 1
100.3.b.c 2 15.d odd 2 1
100.3.b.c 2 60.h even 2 1
180.3.f.a 1 5.c odd 4 1
180.3.f.a 1 20.e even 4 1
180.3.f.b 1 5.c odd 4 1
180.3.f.b 1 20.e even 4 1
320.3.h.a 1 120.q odd 4 1
320.3.h.a 1 120.w even 4 1
320.3.h.b 1 120.q odd 4 1
320.3.h.b 1 120.w even 4 1
900.3.c.h 2 1.a even 1 1 trivial
900.3.c.h 2 4.b odd 2 1 inner
900.3.c.h 2 5.b even 2 1 inner
900.3.c.h 2 20.d odd 2 1 CM
1280.3.e.b 2 240.z odd 4 1
1280.3.e.b 2 240.bb even 4 1
1280.3.e.b 2 240.bd odd 4 1
1280.3.e.b 2 240.bf even 4 1
1280.3.e.c 2 240.z odd 4 1
1280.3.e.c 2 240.bb even 4 1
1280.3.e.c 2 240.bd odd 4 1
1280.3.e.c 2 240.bf even 4 1
1600.3.b.f 2 24.f even 2 1
1600.3.b.f 2 24.h odd 2 1
1600.3.b.f 2 120.i odd 2 1
1600.3.b.f 2 120.m even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(900, [\chi])$$:

 $$T_{7}^{2} + 16$$ T7^2 + 16 $$T_{13}$$ T13 $$T_{17}$$ T17

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 4$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 16$$
$11$ $$T^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2}$$
$19$ $$T^{2}$$
$23$ $$T^{2} + 1936$$
$29$ $$(T + 22)^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2}$$
$41$ $$(T + 62)^{2}$$
$43$ $$T^{2} + 5776$$
$47$ $$T^{2} + 16$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$(T + 58)^{2}$$
$67$ $$T^{2} + 13456$$
$71$ $$T^{2}$$
$73$ $$T^{2}$$
$79$ $$T^{2}$$
$83$ $$T^{2} + 5776$$
$89$ $$(T + 142)^{2}$$
$97$ $$T^{2}$$