# Properties

 Label 2175.1.b.a Level $2175$ Weight $1$ Character orbit 2175.b Analytic conductor $1.085$ Analytic rank $0$ Dimension $2$ Projective image $D_{3}$ CM discriminant -87 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2175,1,Mod(2174,2175)]

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, 1, 1]))

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

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

chi := DirichletCharacter("2175.2174");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2175 = 3 \cdot 5^{2} \cdot 29$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2175.b (of order $$2$$, degree $$1$$, not 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: $$2$$ Coefficient field: $$\Q(i)$$ 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 87) Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.87.1 Artin image: $C_4\times S_3$ Artin field: Galois closure of $$\mathbb{Q}[x]/(x^{12} - \cdots)$$

## $q$-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.

 $$f(q)$$ $$=$$ $$q - i q^{2} - i q^{3} - q^{6} - i q^{7} - i q^{8} - q^{9} +O(q^{10})$$ q - z * q^2 - z * q^3 - q^6 - z * q^7 - z * q^8 - q^9 $$q - i q^{2} - i q^{3} - q^{6} - i q^{7} - i q^{8} - q^{9} - q^{11} + i q^{13} - q^{14} - q^{16} - i q^{17} + i q^{18} - q^{21} + i q^{22} - q^{24} + q^{26} + i q^{27} - q^{29} + i q^{33} - q^{34} + q^{39} + 2 q^{41} + i q^{42} - i q^{47} + i q^{48} - q^{51} + q^{54} - q^{56} + i q^{58} + i q^{63} - q^{64} + q^{66} - i q^{67} + i q^{72} + i q^{77} - i q^{78} + q^{81} - 2 i q^{82} + i q^{87} + i q^{88} + q^{89} + q^{91} - q^{94} + q^{99} +O(q^{100})$$ q - z * q^2 - z * q^3 - q^6 - z * q^7 - z * q^8 - q^9 - q^11 + z * q^13 - q^14 - q^16 - z * q^17 + z * q^18 - q^21 + z * q^22 - q^24 + q^26 + z * q^27 - q^29 + z * q^33 - q^34 + q^39 + 2 * q^41 + z * q^42 - z * q^47 + z * q^48 - q^51 + q^54 - q^56 + z * q^58 + z * q^63 - q^64 + q^66 - z * q^67 + z * q^72 + z * q^77 - z * q^78 + q^81 - 2*z * q^82 + z * q^87 + z * q^88 + q^89 + q^91 - q^94 + q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{6} - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^6 - 2 * q^9 $$2 q - 2 q^{6} - 2 q^{9} - 2 q^{11} - 2 q^{14} - 2 q^{16} - 2 q^{21} - 2 q^{24} + 2 q^{26} - 2 q^{29} - 2 q^{34} + 2 q^{39} + 4 q^{41} - 2 q^{51} + 2 q^{54} - 2 q^{56} - 2 q^{64} + 2 q^{66} + 2 q^{81} + 2 q^{89} + 2 q^{91} - 2 q^{94} + 2 q^{99}+O(q^{100})$$ 2 * q - 2 * q^6 - 2 * q^9 - 2 * q^11 - 2 * q^14 - 2 * q^16 - 2 * q^21 - 2 * q^24 + 2 * q^26 - 2 * q^29 - 2 * q^34 + 2 * q^39 + 4 * q^41 - 2 * q^51 + 2 * q^54 - 2 * q^56 - 2 * q^64 + 2 * q^66 + 2 * q^81 + 2 * q^89 + 2 * q^91 - 2 * q^94 + 2 * q^99

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2174.1
 1.00000i − 1.00000i
1.00000i 1.00000i 0 0 −1.00000 1.00000i 1.00000i −1.00000 0
2174.2 1.00000i 1.00000i 0 0 −1.00000 1.00000i 1.00000i −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 $$\Q(\sqrt{-87})$$
5.b even 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.b.a 2
3.b odd 2 1 2175.1.b.b 2
5.b even 2 1 inner 2175.1.b.a 2
5.c odd 4 1 87.1.d.a 1
5.c odd 4 1 2175.1.h.b 1
15.d odd 2 1 2175.1.b.b 2
15.e even 4 1 87.1.d.b yes 1
15.e even 4 1 2175.1.h.a 1
20.e even 4 1 1392.1.i.a 1
29.b even 2 1 2175.1.b.b 2
45.k odd 12 2 2349.1.h.b 2
45.l even 12 2 2349.1.h.a 2
60.l odd 4 1 1392.1.i.b 1
87.d odd 2 1 CM 2175.1.b.a 2
145.d even 2 1 2175.1.b.b 2
145.e even 4 1 2523.1.b.b 2
145.h odd 4 1 87.1.d.b yes 1
145.h odd 4 1 2175.1.h.a 1
145.j even 4 1 2523.1.b.b 2
145.o even 28 6 2523.1.j.b 12
145.p odd 28 6 2523.1.h.b 6
145.q odd 28 6 2523.1.h.a 6
145.t even 28 6 2523.1.j.b 12
435.b odd 2 1 inner 2175.1.b.a 2
435.i odd 4 1 2523.1.b.b 2
435.p even 4 1 87.1.d.a 1
435.p even 4 1 2175.1.h.b 1
435.t odd 4 1 2523.1.b.b 2
435.bc odd 28 6 2523.1.j.b 12
435.bg even 28 6 2523.1.h.b 6
435.bj even 28 6 2523.1.h.a 6
435.bn odd 28 6 2523.1.j.b 12
580.o even 4 1 1392.1.i.b 1
1305.bi even 12 2 2349.1.h.b 2
1305.bk odd 12 2 2349.1.h.a 2
1740.v odd 4 1 1392.1.i.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
87.1.d.a 1 5.c odd 4 1
87.1.d.a 1 435.p even 4 1
87.1.d.b yes 1 15.e even 4 1
87.1.d.b yes 1 145.h odd 4 1
1392.1.i.a 1 20.e even 4 1
1392.1.i.a 1 1740.v odd 4 1
1392.1.i.b 1 60.l odd 4 1
1392.1.i.b 1 580.o even 4 1
2175.1.b.a 2 1.a even 1 1 trivial
2175.1.b.a 2 5.b even 2 1 inner
2175.1.b.a 2 87.d odd 2 1 CM
2175.1.b.a 2 435.b odd 2 1 inner
2175.1.b.b 2 3.b odd 2 1
2175.1.b.b 2 15.d odd 2 1
2175.1.b.b 2 29.b even 2 1
2175.1.b.b 2 145.d even 2 1
2175.1.h.a 1 15.e even 4 1
2175.1.h.a 1 145.h odd 4 1
2175.1.h.b 1 5.c odd 4 1
2175.1.h.b 1 435.p even 4 1
2349.1.h.a 2 45.l even 12 2
2349.1.h.a 2 1305.bk odd 12 2
2349.1.h.b 2 45.k odd 12 2
2349.1.h.b 2 1305.bi even 12 2
2523.1.b.b 2 145.e even 4 1
2523.1.b.b 2 145.j even 4 1
2523.1.b.b 2 435.i odd 4 1
2523.1.b.b 2 435.t odd 4 1
2523.1.h.a 6 145.q odd 28 6
2523.1.h.a 6 435.bj even 28 6
2523.1.h.b 6 145.p odd 28 6
2523.1.h.b 6 435.bg even 28 6
2523.1.j.b 12 145.o even 28 6
2523.1.j.b 12 145.t even 28 6
2523.1.j.b 12 435.bc odd 28 6
2523.1.j.b 12 435.bn odd 28 6

## Hecke kernels

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

 $$T_{2}^{2} + 1$$ T2^2 + 1 $$T_{11} + 1$$ T11 + 1

## Hecke characteristic polynomials

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