# Properties

 Label 2175.2.c.e Level $2175$ Weight $2$ Character orbit 2175.c Analytic conductor $17.367$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2175,2,Mod(349,2175)]

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([0, 1, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("2175.349");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2175 = 3 \cdot 5^{2} \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2175.c (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: $$17.3674624396$$ 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 435) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + i q^{3} + 2 q^{4} - 2 i q^{7} - q^{9} +O(q^{10})$$ q + i * q^3 + 2 * q^4 - 2*i * q^7 - q^9 $$q + i q^{3} + 2 q^{4} - 2 i q^{7} - q^{9} + 3 q^{11} + 2 i q^{12} + 2 i q^{13} + 4 q^{16} - 2 q^{19} + 2 q^{21} + 3 i q^{23} - i q^{27} - 4 i q^{28} + q^{29} + 8 q^{31} + 3 i q^{33} - 2 q^{36} + i q^{37} - 2 q^{39} - 3 q^{41} - i q^{43} + 6 q^{44} + 6 i q^{47} + 4 i q^{48} + 3 q^{49} + 4 i q^{52} - 3 i q^{53} - 2 i q^{57} + 12 q^{59} + 8 q^{61} + 2 i q^{63} + 8 q^{64} - 14 i q^{67} - 3 q^{69} - 6 q^{71} - 7 i q^{73} - 4 q^{76} - 6 i q^{77} + 4 q^{79} + q^{81} + 9 i q^{83} + 4 q^{84} + i q^{87} + 6 q^{89} + 4 q^{91} + 6 i q^{92} + 8 i q^{93} - 11 i q^{97} - 3 q^{99} +O(q^{100})$$ q + i * q^3 + 2 * q^4 - 2*i * q^7 - q^9 + 3 * q^11 + 2*i * q^12 + 2*i * q^13 + 4 * q^16 - 2 * q^19 + 2 * q^21 + 3*i * q^23 - i * q^27 - 4*i * q^28 + q^29 + 8 * q^31 + 3*i * q^33 - 2 * q^36 + i * q^37 - 2 * q^39 - 3 * q^41 - i * q^43 + 6 * q^44 + 6*i * q^47 + 4*i * q^48 + 3 * q^49 + 4*i * q^52 - 3*i * q^53 - 2*i * q^57 + 12 * q^59 + 8 * q^61 + 2*i * q^63 + 8 * q^64 - 14*i * q^67 - 3 * q^69 - 6 * q^71 - 7*i * q^73 - 4 * q^76 - 6*i * q^77 + 4 * q^79 + q^81 + 9*i * q^83 + 4 * q^84 + i * q^87 + 6 * q^89 + 4 * q^91 + 6*i * q^92 + 8*i * q^93 - 11*i * q^97 - 3 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{4} - 2 q^{9}+O(q^{10})$$ 2 * q + 4 * q^4 - 2 * q^9 $$2 q + 4 q^{4} - 2 q^{9} + 6 q^{11} + 8 q^{16} - 4 q^{19} + 4 q^{21} + 2 q^{29} + 16 q^{31} - 4 q^{36} - 4 q^{39} - 6 q^{41} + 12 q^{44} + 6 q^{49} + 24 q^{59} + 16 q^{61} + 16 q^{64} - 6 q^{69} - 12 q^{71} - 8 q^{76} + 8 q^{79} + 2 q^{81} + 8 q^{84} + 12 q^{89} + 8 q^{91} - 6 q^{99}+O(q^{100})$$ 2 * q + 4 * q^4 - 2 * q^9 + 6 * q^11 + 8 * q^16 - 4 * q^19 + 4 * q^21 + 2 * q^29 + 16 * q^31 - 4 * q^36 - 4 * q^39 - 6 * q^41 + 12 * q^44 + 6 * q^49 + 24 * q^59 + 16 * q^61 + 16 * q^64 - 6 * q^69 - 12 * q^71 - 8 * q^76 + 8 * q^79 + 2 * q^81 + 8 * q^84 + 12 * q^89 + 8 * q^91 - 6 * 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
349.1
 − 1.00000i 1.00000i
0 1.00000i 2.00000 0 0 2.00000i 0 −1.00000 0
349.2 0 1.00000i 2.00000 0 0 2.00000i 0 −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
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 2175.2.c.e 2
5.b even 2 1 inner 2175.2.c.e 2
5.c odd 4 1 435.2.a.c 1
5.c odd 4 1 2175.2.a.d 1
15.e even 4 1 1305.2.a.d 1
15.e even 4 1 6525.2.a.f 1
20.e even 4 1 6960.2.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
435.2.a.c 1 5.c odd 4 1
1305.2.a.d 1 15.e even 4 1
2175.2.a.d 1 5.c odd 4 1
2175.2.c.e 2 1.a even 1 1 trivial
2175.2.c.e 2 5.b even 2 1 inner
6525.2.a.f 1 15.e even 4 1
6960.2.a.b 1 20.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2175, [\chi])$$:

 $$T_{2}$$ T2 $$T_{7}^{2} + 4$$ T7^2 + 4 $$T_{11} - 3$$ T11 - 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 1$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 4$$
$11$ $$(T - 3)^{2}$$
$13$ $$T^{2} + 4$$
$17$ $$T^{2}$$
$19$ $$(T + 2)^{2}$$
$23$ $$T^{2} + 9$$
$29$ $$(T - 1)^{2}$$
$31$ $$(T - 8)^{2}$$
$37$ $$T^{2} + 1$$
$41$ $$(T + 3)^{2}$$
$43$ $$T^{2} + 1$$
$47$ $$T^{2} + 36$$
$53$ $$T^{2} + 9$$
$59$ $$(T - 12)^{2}$$
$61$ $$(T - 8)^{2}$$
$67$ $$T^{2} + 196$$
$71$ $$(T + 6)^{2}$$
$73$ $$T^{2} + 49$$
$79$ $$(T - 4)^{2}$$
$83$ $$T^{2} + 81$$
$89$ $$(T - 6)^{2}$$
$97$ $$T^{2} + 121$$