# Properties

 Label 2175.1.h.d Level $2175$ Weight $1$ Character orbit 2175.h Self dual yes Analytic conductor $1.085$ Analytic rank $0$ Dimension $3$ Projective image $D_{9}$ CM discriminant -87 Inner twists $2$

# 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: yes Analytic conductor: $$1.08546640248$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: $$\Q(\zeta_{18})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - 3x - 1$$ x^3 - 3*x - 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{9}$$ Projective field: Galois closure of 9.1.895152515625.1

## $q$-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 $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{2} - \beta_1) q^{2} - q^{3} + ( - \beta_1 + 1) q^{4} + ( - \beta_{2} + \beta_1) q^{6} - \beta_1 q^{7} + (\beta_{2} - \beta_1 + 1) q^{8} + q^{9}+O(q^{10})$$ q + (b2 - b1) * q^2 - q^3 + (-b1 + 1) * q^4 + (-b2 + b1) * q^6 - b1 * q^7 + (b2 - b1 + 1) * q^8 + q^9 $$q + (\beta_{2} - \beta_1) q^{2} - q^{3} + ( - \beta_1 + 1) q^{4} + ( - \beta_{2} + \beta_1) q^{6} - \beta_1 q^{7} + (\beta_{2} - \beta_1 + 1) q^{8} + q^{9} - \beta_{2} q^{11} + (\beta_1 - 1) q^{12} + ( - \beta_{2} + \beta_1) q^{13} + (\beta_{2} - \beta_1 + 1) q^{14} + (\beta_{2} - \beta_1 + 1) q^{16} - \beta_{2} q^{17} + (\beta_{2} - \beta_1) q^{18} + \beta_1 q^{21} + (\beta_{2} - 1) q^{22} + ( - \beta_{2} + \beta_1 - 1) q^{24} + (\beta_1 - 2) q^{26} - q^{27} + (\beta_{2} - \beta_1 + 2) q^{28} - q^{29} + ( - \beta_1 + 1) q^{32} + \beta_{2} q^{33} + (\beta_{2} - 1) q^{34} + ( - \beta_1 + 1) q^{36} + (\beta_{2} - \beta_1) q^{39} + q^{41} + ( - \beta_{2} + \beta_1 - 1) q^{42} + ( - \beta_{2} + \beta_1 + 1) q^{44} + \beta_1 q^{47} + ( - \beta_{2} + \beta_1 - 1) q^{48} + (\beta_{2} + 1) q^{49} + \beta_{2} q^{51} + ( - 2 \beta_{2} + 2 \beta_1 - 1) q^{52} + ( - \beta_{2} + \beta_1) q^{54} + (\beta_{2} - 2 \beta_1 + 1) q^{56} + ( - \beta_{2} + \beta_1) q^{58} - \beta_1 q^{63} + (\beta_{2} - \beta_1) q^{64} + ( - \beta_{2} + 1) q^{66} + \beta_{2} q^{67} + ( - \beta_{2} + \beta_1 + 1) q^{68} + (\beta_{2} - \beta_1 + 1) q^{72} + (\beta_1 + 1) q^{77} + ( - \beta_1 + 2) q^{78} + q^{81} + (\beta_{2} - \beta_1) q^{82} + ( - \beta_{2} + \beta_1 - 2) q^{84} + q^{87} - q^{88} + (\beta_{2} - \beta_1) q^{89} + ( - \beta_{2} + \beta_1 - 1) q^{91} + ( - \beta_{2} + \beta_1 - 1) q^{94} + (\beta_1 - 1) q^{96} + ( - \beta_1 + 1) q^{98} - \beta_{2} q^{99}+O(q^{100})$$ q + (b2 - b1) * q^2 - q^3 + (-b1 + 1) * q^4 + (-b2 + b1) * q^6 - b1 * q^7 + (b2 - b1 + 1) * q^8 + q^9 - b2 * q^11 + (b1 - 1) * q^12 + (-b2 + b1) * q^13 + (b2 - b1 + 1) * q^14 + (b2 - b1 + 1) * q^16 - b2 * q^17 + (b2 - b1) * q^18 + b1 * q^21 + (b2 - 1) * q^22 + (-b2 + b1 - 1) * q^24 + (b1 - 2) * q^26 - q^27 + (b2 - b1 + 2) * q^28 - q^29 + (-b1 + 1) * q^32 + b2 * q^33 + (b2 - 1) * q^34 + (-b1 + 1) * q^36 + (b2 - b1) * q^39 + q^41 + (-b2 + b1 - 1) * q^42 + (-b2 + b1 + 1) * q^44 + b1 * q^47 + (-b2 + b1 - 1) * q^48 + (b2 + 1) * q^49 + b2 * q^51 + (-2*b2 + 2*b1 - 1) * q^52 + (-b2 + b1) * q^54 + (b2 - 2*b1 + 1) * q^56 + (-b2 + b1) * q^58 - b1 * q^63 + (b2 - b1) * q^64 + (-b2 + 1) * q^66 + b2 * q^67 + (-b2 + b1 + 1) * q^68 + (b2 - b1 + 1) * q^72 + (b1 + 1) * q^77 + (-b1 + 2) * q^78 + q^81 + (b2 - b1) * q^82 + (-b2 + b1 - 2) * q^84 + q^87 - q^88 + (b2 - b1) * q^89 + (-b2 + b1 - 1) * q^91 + (-b2 + b1 - 1) * q^94 + (b1 - 1) * q^96 + (-b1 + 1) * q^98 - b2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{3} + 3 q^{4} + 3 q^{8} + 3 q^{9}+O(q^{10})$$ 3 * q - 3 * q^3 + 3 * q^4 + 3 * q^8 + 3 * q^9 $$3 q - 3 q^{3} + 3 q^{4} + 3 q^{8} + 3 q^{9} - 3 q^{12} + 3 q^{14} + 3 q^{16} - 3 q^{22} - 3 q^{24} - 6 q^{26} - 3 q^{27} + 6 q^{28} - 3 q^{29} + 3 q^{32} - 3 q^{34} + 3 q^{36} + 3 q^{41} - 3 q^{42} + 3 q^{44} - 3 q^{48} + 3 q^{49} - 3 q^{52} + 3 q^{56} + 3 q^{66} + 3 q^{68} + 3 q^{72} + 3 q^{77} + 6 q^{78} + 3 q^{81} - 6 q^{84} + 3 q^{87} - 3 q^{88} - 3 q^{91} - 3 q^{94} - 3 q^{96} + 3 q^{98}+O(q^{100})$$ 3 * q - 3 * q^3 + 3 * q^4 + 3 * q^8 + 3 * q^9 - 3 * q^12 + 3 * q^14 + 3 * q^16 - 3 * q^22 - 3 * q^24 - 6 * q^26 - 3 * q^27 + 6 * q^28 - 3 * q^29 + 3 * q^32 - 3 * q^34 + 3 * q^36 + 3 * q^41 - 3 * q^42 + 3 * q^44 - 3 * q^48 + 3 * q^49 - 3 * q^52 + 3 * q^56 + 3 * q^66 + 3 * q^68 + 3 * q^72 + 3 * q^77 + 6 * q^78 + 3 * q^81 - 6 * q^84 + 3 * q^87 - 3 * q^88 - 3 * q^91 - 3 * q^94 - 3 * q^96 + 3 * q^98

Basis of coefficient ring in terms of $$\nu = \zeta_{18} + \zeta_{18}^{-1}$$:

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2$$ v^2 - 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2$$ b2 + 2

## 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
 −0.347296 1.87939 −1.53209
−1.53209 −1.00000 1.34730 0 1.53209 0.347296 −0.532089 1.00000 0
1826.2 −0.347296 −1.00000 −0.879385 0 0.347296 −1.87939 0.652704 1.00000 0
1826.3 1.87939 −1.00000 2.53209 0 −1.87939 1.53209 2.87939 1.00000 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
87.d odd 2 1 CM by $$\Q(\sqrt{-87})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2175.1.h.d yes 3
3.b odd 2 1 2175.1.h.e yes 3
5.b even 2 1 2175.1.h.f yes 3
5.c odd 4 2 2175.1.b.c 6
15.d odd 2 1 2175.1.h.c 3
15.e even 4 2 2175.1.b.d 6
29.b even 2 1 2175.1.h.e yes 3
87.d odd 2 1 CM 2175.1.h.d yes 3
145.d even 2 1 2175.1.h.c 3
145.h odd 4 2 2175.1.b.d 6
435.b odd 2 1 2175.1.h.f yes 3
435.p even 4 2 2175.1.b.c 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2175.1.b.c 6 5.c odd 4 2
2175.1.b.c 6 435.p even 4 2
2175.1.b.d 6 15.e even 4 2
2175.1.b.d 6 145.h odd 4 2
2175.1.h.c 3 15.d odd 2 1
2175.1.h.c 3 145.d even 2 1
2175.1.h.d yes 3 1.a even 1 1 trivial
2175.1.h.d yes 3 87.d odd 2 1 CM
2175.1.h.e yes 3 3.b odd 2 1
2175.1.h.e yes 3 29.b even 2 1
2175.1.h.f yes 3 5.b even 2 1
2175.1.h.f yes 3 435.b odd 2 1

## Hecke kernels

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

 $$T_{2}^{3} - 3T_{2} - 1$$ T2^3 - 3*T2 - 1 $$T_{7}^{3} - 3T_{7} + 1$$ T7^3 - 3*T7 + 1

## Hecke characteristic polynomials

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