# Properties

 Label 2175.1.b.c Level $2175$ Weight $1$ Character orbit 2175.b Analytic conductor $1.085$ Analytic rank $0$ Dimension $6$ Projective image $D_{9}$ 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: $$6$$ Coefficient field: 6.0.419904.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} + 6x^{4} + 9x^{2} + 1$$ x^6 + 6*x^4 + 9*x^2 + 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{5} q^{2} - \beta_{3} q^{3} + (\beta_{4} - \beta_{2} - 1) q^{4} + \beta_{4} q^{6} - \beta_1 q^{7} + ( - \beta_{5} + \beta_{3}) q^{8} - q^{9}+O(q^{10})$$ q + b5 * q^2 - b3 * q^3 + (b4 - b2 - 1) * q^4 + b4 * q^6 - b1 * q^7 + (-b5 + b3) * q^8 - q^9 $$q + \beta_{5} q^{2} - \beta_{3} q^{3} + (\beta_{4} - \beta_{2} - 1) q^{4} + \beta_{4} q^{6} - \beta_1 q^{7} + ( - \beta_{5} + \beta_{3}) q^{8} - q^{9} + \beta_{2} q^{11} + (\beta_{3} + \beta_1) q^{12} + \beta_{5} q^{13} + (\beta_{4} - 1) q^{14} + ( - \beta_{4} + 1) q^{16} + ( - \beta_{5} - \beta_1) q^{17} - \beta_{5} q^{18} + (\beta_{4} - \beta_{2}) q^{21} + (\beta_{5} + \beta_{3} + \beta_1) q^{22} + ( - \beta_{4} + 1) q^{24} + (\beta_{4} - \beta_{2} - 2) q^{26} + \beta_{3} q^{27} + ( - \beta_{5} + 2 \beta_{3}) q^{28} + q^{29} + ( - \beta_{3} - \beta_1) q^{32} + ( - \beta_{5} - \beta_1) q^{33} + (\beta_{2} + 1) q^{34} + ( - \beta_{4} + \beta_{2} + 1) q^{36} + \beta_{4} q^{39} + q^{41} + ( - \beta_{5} + \beta_{3}) q^{42} + ( - \beta_{4} - 1) q^{44} + \beta_1 q^{47} + (\beta_{5} - \beta_{3}) q^{48} + (\beta_{2} - 1) q^{49} - \beta_{2} q^{51} + ( - 2 \beta_{5} + \beta_{3}) q^{52} - \beta_{4} q^{54} + ( - 2 \beta_{4} + \beta_{2} + 1) q^{56} + \beta_{5} q^{58} + \beta_1 q^{63} + \beta_{4} q^{64} + (\beta_{2} + 1) q^{66} + (\beta_{5} + \beta_1) q^{67} + (\beta_{5} + \beta_{3}) q^{68} + (\beta_{5} - \beta_{3}) q^{72} + ( - \beta_{3} + \beta_1) q^{77} + (2 \beta_{3} + \beta_1) q^{78} + q^{81} + \beta_{5} q^{82} + ( - \beta_{4} + 2) q^{84} - \beta_{3} q^{87} - \beta_{3} q^{88} + \beta_{4} q^{89} + (\beta_{4} - 1) q^{91} + ( - \beta_{4} + 1) q^{94} + (\beta_{4} - \beta_{2} - 1) q^{96} + (\beta_{3} + \beta_1) q^{98} - \beta_{2} q^{99}+O(q^{100})$$ q + b5 * q^2 - b3 * q^3 + (b4 - b2 - 1) * q^4 + b4 * q^6 - b1 * q^7 + (-b5 + b3) * q^8 - q^9 + b2 * q^11 + (b3 + b1) * q^12 + b5 * q^13 + (b4 - 1) * q^14 + (-b4 + 1) * q^16 + (-b5 - b1) * q^17 - b5 * q^18 + (b4 - b2) * q^21 + (b5 + b3 + b1) * q^22 + (-b4 + 1) * q^24 + (b4 - b2 - 2) * q^26 + b3 * q^27 + (-b5 + 2*b3) * q^28 + q^29 + (-b3 - b1) * q^32 + (-b5 - b1) * q^33 + (b2 + 1) * q^34 + (-b4 + b2 + 1) * q^36 + b4 * q^39 + q^41 + (-b5 + b3) * q^42 + (-b4 - 1) * q^44 + b1 * q^47 + (b5 - b3) * q^48 + (b2 - 1) * q^49 - b2 * q^51 + (-2*b5 + b3) * q^52 - b4 * q^54 + (-2*b4 + b2 + 1) * q^56 + b5 * q^58 + b1 * q^63 + b4 * q^64 + (b2 + 1) * q^66 + (b5 + b1) * q^67 + (b5 + b3) * q^68 + (b5 - b3) * q^72 + (-b3 + b1) * q^77 + (2*b3 + b1) * q^78 + q^81 + b5 * q^82 + (-b4 + 2) * q^84 - b3 * q^87 - b3 * q^88 + b4 * q^89 + (b4 - 1) * q^91 + (-b4 + 1) * q^94 + (b4 - b2 - 1) * q^96 + (b3 + b1) * q^98 - b2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 6 q^{4} - 6 q^{9}+O(q^{10})$$ 6 * q - 6 * q^4 - 6 * q^9 $$6 q - 6 q^{4} - 6 q^{9} - 6 q^{14} + 6 q^{16} + 6 q^{24} - 12 q^{26} + 6 q^{29} + 6 q^{34} + 6 q^{36} + 6 q^{41} - 6 q^{44} - 6 q^{49} + 6 q^{56} + 6 q^{66} + 6 q^{81} + 12 q^{84} - 6 q^{91} + 6 q^{94} - 6 q^{96}+O(q^{100})$$ 6 * q - 6 * q^4 - 6 * q^9 - 6 * q^14 + 6 * q^16 + 6 * q^24 - 12 * q^26 + 6 * q^29 + 6 * q^34 + 6 * q^36 + 6 * q^41 - 6 * q^44 - 6 * q^49 + 6 * q^56 + 6 * q^66 + 6 * q^81 + 12 * q^84 - 6 * q^91 + 6 * q^94 - 6 * q^96

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} + 6x^{4} + 9x^{2} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} + 2$$ v^2 + 2 $$\beta_{3}$$ $$=$$ $$\nu^{3} + 3\nu$$ v^3 + 3*v $$\beta_{4}$$ $$=$$ $$\nu^{4} + 4\nu^{2} + 2$$ v^4 + 4*v^2 + 2 $$\beta_{5}$$ $$=$$ $$\nu^{5} + 5\nu^{3} + 5\nu$$ v^5 + 5*v^3 + 5*v
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} - 2$$ b2 - 2 $$\nu^{3}$$ $$=$$ $$\beta_{3} - 3\beta_1$$ b3 - 3*b1 $$\nu^{4}$$ $$=$$ $$\beta_{4} - 4\beta_{2} + 6$$ b4 - 4*b2 + 6 $$\nu^{5}$$ $$=$$ $$\beta_{5} - 5\beta_{3} + 10\beta_1$$ b5 - 5*b3 + 10*b1

## 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
 1.53209i − 0.347296i 1.87939i − 1.87939i 0.347296i − 1.53209i
1.87939i 1.00000i −2.53209 0 −1.87939 1.53209i 2.87939i −1.00000 0
2174.2 1.53209i 1.00000i −1.34730 0 1.53209 0.347296i 0.532089i −1.00000 0
2174.3 0.347296i 1.00000i 0.879385 0 0.347296 1.87939i 0.652704i −1.00000 0
2174.4 0.347296i 1.00000i 0.879385 0 0.347296 1.87939i 0.652704i −1.00000 0
2174.5 1.53209i 1.00000i −1.34730 0 1.53209 0.347296i 0.532089i −1.00000 0
2174.6 1.87939i 1.00000i −2.53209 0 −1.87939 1.53209i 2.87939i −1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2174.6 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.c 6
3.b odd 2 1 2175.1.b.d 6
5.b even 2 1 inner 2175.1.b.c 6
5.c odd 4 1 2175.1.h.d yes 3
5.c odd 4 1 2175.1.h.f yes 3
15.d odd 2 1 2175.1.b.d 6
15.e even 4 1 2175.1.h.c 3
15.e even 4 1 2175.1.h.e yes 3
29.b even 2 1 2175.1.b.d 6
87.d odd 2 1 CM 2175.1.b.c 6
145.d even 2 1 2175.1.b.d 6
145.h odd 4 1 2175.1.h.c 3
145.h odd 4 1 2175.1.h.e yes 3
435.b odd 2 1 inner 2175.1.b.c 6
435.p even 4 1 2175.1.h.d yes 3
435.p even 4 1 2175.1.h.f yes 3

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

## Hecke kernels

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

 $$T_{2}^{6} + 6T_{2}^{4} + 9T_{2}^{2} + 1$$ T2^6 + 6*T2^4 + 9*T2^2 + 1 $$T_{11}^{3} - 3T_{11} - 1$$ T11^3 - 3*T11 - 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} + 6 T^{4} + \cdots + 1$$
$3$ $$(T^{2} + 1)^{3}$$
$5$ $$T^{6}$$
$7$ $$T^{6} + 6 T^{4} + \cdots + 1$$
$11$ $$(T^{3} - 3 T - 1)^{2}$$
$13$ $$T^{6} + 6 T^{4} + \cdots + 1$$
$17$ $$T^{6} + 6 T^{4} + \cdots + 1$$
$19$ $$T^{6}$$
$23$ $$T^{6}$$
$29$ $$(T - 1)^{6}$$
$31$ $$T^{6}$$
$37$ $$T^{6}$$
$41$ $$(T - 1)^{6}$$
$43$ $$T^{6}$$
$47$ $$T^{6} + 6 T^{4} + \cdots + 1$$
$53$ $$T^{6}$$
$59$ $$T^{6}$$
$61$ $$T^{6}$$
$67$ $$T^{6} + 6 T^{4} + \cdots + 1$$
$71$ $$T^{6}$$
$73$ $$T^{6}$$
$79$ $$T^{6}$$
$83$ $$T^{6}$$
$89$ $$(T^{3} - 3 T + 1)^{2}$$
$97$ $$T^{6}$$