# Properties

 Label 2175.1.h.g Level $2175$ Weight $1$ Character orbit 2175.h Analytic conductor $1.085$ Analytic rank $0$ Dimension $4$ Projective image $D_{4}$ RM discriminant 145 Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2175,1,Mod(1826,2175)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2175, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("2175.1826");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2175 = 3 \cdot 5^{2} \cdot 29$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2175.h (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: $$1.08546640248$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 435) Projective image: $$D_{4}$$ Projective field: Galois closure of 4.0.6525.1

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q + (\zeta_{8}^{3} - \zeta_{8}) q^{2} - \zeta_{8} q^{3} + q^{4} + (\zeta_{8}^{2} + 1) q^{6} + \zeta_{8}^{2} q^{9} +O(q^{10})$$ q + (z^3 - z) * q^2 - z * q^3 + q^4 + (z^2 + 1) * q^6 + z^2 * q^9 $$q + (\zeta_{8}^{3} - \zeta_{8}) q^{2} - \zeta_{8} q^{3} + q^{4} + (\zeta_{8}^{2} + 1) q^{6} + \zeta_{8}^{2} q^{9} - \zeta_{8} q^{12} - q^{16} + (\zeta_{8}^{3} - \zeta_{8}) q^{17} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{18} - \zeta_{8}^{3} q^{27} - \zeta_{8}^{2} q^{29} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{32} + 2 q^{34} + \zeta_{8}^{2} q^{36} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{37} + (\zeta_{8}^{3} + \zeta_{8}) q^{43} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{47} + \zeta_{8} q^{48} - q^{49} + (\zeta_{8}^{2} + 1) q^{51} + (\zeta_{8}^{2} - 1) q^{54} + (\zeta_{8}^{3} + \zeta_{8}) q^{58} - 2 \zeta_{8}^{2} q^{59} - q^{64} + (\zeta_{8}^{3} - \zeta_{8}) q^{68} - 2 \zeta_{8}^{2} q^{71} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{73} + 2 \zeta_{8}^{2} q^{74} - q^{81} - 2 \zeta_{8}^{2} q^{86} + \zeta_{8}^{3} q^{87} - 2 q^{94} + ( - \zeta_{8}^{2} - 1) q^{96} + (\zeta_{8}^{3} + \zeta_{8}) q^{97} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{98} +O(q^{100})$$ q + (z^3 - z) * q^2 - z * q^3 + q^4 + (z^2 + 1) * q^6 + z^2 * q^9 - z * q^12 - q^16 + (z^3 - z) * q^17 + (-z^3 - z) * q^18 - z^3 * q^27 - z^2 * q^29 + (-z^3 + z) * q^32 + 2 * q^34 + z^2 * q^36 + (-z^3 - z) * q^37 + (z^3 + z) * q^43 + (-z^3 + z) * q^47 + z * q^48 - q^49 + (z^2 + 1) * q^51 + (z^2 - 1) * q^54 + (z^3 + z) * q^58 - 2*z^2 * q^59 - q^64 + (z^3 - z) * q^68 - 2*z^2 * q^71 + (-z^3 - z) * q^73 + 2*z^2 * q^74 - q^81 - 2*z^2 * q^86 + z^3 * q^87 - 2 * q^94 + (-z^2 - 1) * q^96 + (z^3 + z) * q^97 + (-z^3 + z) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{4} + 4 q^{6}+O(q^{10})$$ 4 * q + 4 * q^4 + 4 * q^6 $$4 q + 4 q^{4} + 4 q^{6} - 4 q^{16} + 8 q^{34} - 4 q^{49} + 4 q^{51} - 4 q^{54} - 4 q^{64} - 4 q^{81} - 8 q^{94} - 4 q^{96}+O(q^{100})$$ 4 * q + 4 * q^4 + 4 * q^6 - 4 * q^16 + 8 * q^34 - 4 * q^49 + 4 * q^51 - 4 * q^54 - 4 * q^64 - 4 * q^81 - 8 * q^94 - 4 * q^96

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2175\mathbb{Z}\right)^\times$$.

 $$n$$ $$901$$ $$1451$$ $$2002$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1826.1
 0.707107 + 0.707107i 0.707107 − 0.707107i −0.707107 + 0.707107i −0.707107 − 0.707107i
−1.41421 −0.707107 0.707107i 1.00000 0 1.00000 + 1.00000i 0 0 1.00000i 0
1826.2 −1.41421 −0.707107 + 0.707107i 1.00000 0 1.00000 1.00000i 0 0 1.00000i 0
1826.3 1.41421 0.707107 0.707107i 1.00000 0 1.00000 1.00000i 0 0 1.00000i 0
1826.4 1.41421 0.707107 + 0.707107i 1.00000 0 1.00000 + 1.00000i 0 0 1.00000i 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
145.d even 2 1 RM by $$\Q(\sqrt{145})$$
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner
29.b even 2 1 inner
87.d odd 2 1 inner
435.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2175.1.h.g 4
3.b odd 2 1 inner 2175.1.h.g 4
5.b even 2 1 inner 2175.1.h.g 4
5.c odd 4 2 435.1.b.a 4
15.d odd 2 1 inner 2175.1.h.g 4
15.e even 4 2 435.1.b.a 4
29.b even 2 1 inner 2175.1.h.g 4
87.d odd 2 1 inner 2175.1.h.g 4
145.d even 2 1 RM 2175.1.h.g 4
145.h odd 4 2 435.1.b.a 4
435.b odd 2 1 inner 2175.1.h.g 4
435.p even 4 2 435.1.b.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
435.1.b.a 4 5.c odd 4 2
435.1.b.a 4 15.e even 4 2
435.1.b.a 4 145.h odd 4 2
435.1.b.a 4 435.p even 4 2
2175.1.h.g 4 1.a even 1 1 trivial
2175.1.h.g 4 3.b odd 2 1 inner
2175.1.h.g 4 5.b even 2 1 inner
2175.1.h.g 4 15.d odd 2 1 inner
2175.1.h.g 4 29.b even 2 1 inner
2175.1.h.g 4 87.d odd 2 1 inner
2175.1.h.g 4 145.d even 2 1 RM
2175.1.h.g 4 435.b odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(2175, [\chi])$$:

 $$T_{2}^{2} - 2$$ T2^2 - 2 $$T_{7}$$ T7

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - 2)^{2}$$
$3$ $$T^{4} + 1$$
$5$ $$T^{4}$$
$7$ $$T^{4}$$
$11$ $$T^{4}$$
$13$ $$T^{4}$$
$17$ $$(T^{2} - 2)^{2}$$
$19$ $$T^{4}$$
$23$ $$T^{4}$$
$29$ $$(T^{2} + 1)^{2}$$
$31$ $$T^{4}$$
$37$ $$(T^{2} + 2)^{2}$$
$41$ $$T^{4}$$
$43$ $$(T^{2} + 2)^{2}$$
$47$ $$(T^{2} - 2)^{2}$$
$53$ $$T^{4}$$
$59$ $$(T^{2} + 4)^{2}$$
$61$ $$T^{4}$$
$67$ $$T^{4}$$
$71$ $$(T^{2} + 4)^{2}$$
$73$ $$(T^{2} + 2)^{2}$$
$79$ $$T^{4}$$
$83$ $$T^{4}$$
$89$ $$T^{4}$$
$97$ $$(T^{2} + 2)^{2}$$