# Properties

 Label 2175.2.a.a Level $2175$ Weight $2$ Character orbit 2175.a Self dual yes Analytic conductor $17.367$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2175,2,Mod(1,2175)]

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, 0, 0]))

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

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

chi := DirichletCharacter("2175.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2175 = 3 \cdot 5^{2} \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2175.a (trivial)

## 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: $$17.3674624396$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 435) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q - 2 q^{2} + q^{3} + 2 q^{4} - 2 q^{6} - 2 q^{7} + q^{9}+O(q^{10})$$ q - 2 * q^2 + q^3 + 2 * q^4 - 2 * q^6 - 2 * q^7 + q^9 $$q - 2 q^{2} + q^{3} + 2 q^{4} - 2 q^{6} - 2 q^{7} + q^{9} - 3 q^{11} + 2 q^{12} + 4 q^{13} + 4 q^{14} - 4 q^{16} + 8 q^{17} - 2 q^{18} - 2 q^{21} + 6 q^{22} - q^{23} - 8 q^{26} + q^{27} - 4 q^{28} + q^{29} - 8 q^{31} + 8 q^{32} - 3 q^{33} - 16 q^{34} + 2 q^{36} - 7 q^{37} + 4 q^{39} + 7 q^{41} + 4 q^{42} + 9 q^{43} - 6 q^{44} + 2 q^{46} - 12 q^{47} - 4 q^{48} - 3 q^{49} + 8 q^{51} + 8 q^{52} + 9 q^{53} - 2 q^{54} - 2 q^{58} + 10 q^{59} + 2 q^{61} + 16 q^{62} - 2 q^{63} - 8 q^{64} + 6 q^{66} + 8 q^{67} + 16 q^{68} - q^{69} - 8 q^{71} - q^{73} + 14 q^{74} + 6 q^{77} - 8 q^{78} - 10 q^{79} + q^{81} - 14 q^{82} + 9 q^{83} - 4 q^{84} - 18 q^{86} + q^{87} + 10 q^{89} - 8 q^{91} - 2 q^{92} - 8 q^{93} + 24 q^{94} + 8 q^{96} + 13 q^{97} + 6 q^{98} - 3 q^{99}+O(q^{100})$$ q - 2 * q^2 + q^3 + 2 * q^4 - 2 * q^6 - 2 * q^7 + q^9 - 3 * q^11 + 2 * q^12 + 4 * q^13 + 4 * q^14 - 4 * q^16 + 8 * q^17 - 2 * q^18 - 2 * q^21 + 6 * q^22 - q^23 - 8 * q^26 + q^27 - 4 * q^28 + q^29 - 8 * q^31 + 8 * q^32 - 3 * q^33 - 16 * q^34 + 2 * q^36 - 7 * q^37 + 4 * q^39 + 7 * q^41 + 4 * q^42 + 9 * q^43 - 6 * q^44 + 2 * q^46 - 12 * q^47 - 4 * q^48 - 3 * q^49 + 8 * q^51 + 8 * q^52 + 9 * q^53 - 2 * q^54 - 2 * q^58 + 10 * q^59 + 2 * q^61 + 16 * q^62 - 2 * q^63 - 8 * q^64 + 6 * q^66 + 8 * q^67 + 16 * q^68 - q^69 - 8 * q^71 - q^73 + 14 * q^74 + 6 * q^77 - 8 * q^78 - 10 * q^79 + q^81 - 14 * q^82 + 9 * q^83 - 4 * q^84 - 18 * q^86 + q^87 + 10 * q^89 - 8 * q^91 - 2 * q^92 - 8 * q^93 + 24 * q^94 + 8 * q^96 + 13 * q^97 + 6 * q^98 - 3 * q^99

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
−2.00000 1.00000 2.00000 0 −2.00000 −2.00000 0 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$5$$ $$-1$$
$$29$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2175.2.a.a 1
3.b odd 2 1 6525.2.a.l 1
5.b even 2 1 2175.2.a.j 1
5.c odd 4 2 435.2.c.a 2
15.d odd 2 1 6525.2.a.b 1
15.e even 4 2 1305.2.c.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
435.2.c.a 2 5.c odd 4 2
1305.2.c.a 2 15.e even 4 2
2175.2.a.a 1 1.a even 1 1 trivial
2175.2.a.j 1 5.b even 2 1
6525.2.a.b 1 15.d odd 2 1
6525.2.a.l 1 3.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2175))$$:

 $$T_{2} + 2$$ T2 + 2 $$T_{7} + 2$$ T7 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 2$$
$3$ $$T - 1$$
$5$ $$T$$
$7$ $$T + 2$$
$11$ $$T + 3$$
$13$ $$T - 4$$
$17$ $$T - 8$$
$19$ $$T$$
$23$ $$T + 1$$
$29$ $$T - 1$$
$31$ $$T + 8$$
$37$ $$T + 7$$
$41$ $$T - 7$$
$43$ $$T - 9$$
$47$ $$T + 12$$
$53$ $$T - 9$$
$59$ $$T - 10$$
$61$ $$T - 2$$
$67$ $$T - 8$$
$71$ $$T + 8$$
$73$ $$T + 1$$
$79$ $$T + 10$$
$83$ $$T - 9$$
$89$ $$T - 10$$
$97$ $$T - 13$$