# Properties

 Label 2175.2.a.b Level $2175$ Weight $2$ Character orbit 2175.a Self dual yes Analytic conductor $17.367$ Analytic rank $2$ 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: $$2$$ 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)

 $$f(q)$$ $$=$$ $$q - q^{2} - q^{3} - q^{4} + q^{6} - 4 q^{7} + 3 q^{8} + q^{9}+O(q^{10})$$ q - q^2 - q^3 - q^4 + q^6 - 4 * q^7 + 3 * q^8 + q^9 $$q - q^{2} - q^{3} - q^{4} + q^{6} - 4 q^{7} + 3 q^{8} + q^{9} - 4 q^{11} + q^{12} - 6 q^{13} + 4 q^{14} - q^{16} - 6 q^{17} - q^{18} - 4 q^{19} + 4 q^{21} + 4 q^{22} + 4 q^{23} - 3 q^{24} + 6 q^{26} - q^{27} + 4 q^{28} + q^{29} - 8 q^{31} - 5 q^{32} + 4 q^{33} + 6 q^{34} - q^{36} - 2 q^{37} + 4 q^{38} + 6 q^{39} - 6 q^{41} - 4 q^{42} - 4 q^{43} + 4 q^{44} - 4 q^{46} + q^{48} + 9 q^{49} + 6 q^{51} + 6 q^{52} + 10 q^{53} + q^{54} - 12 q^{56} + 4 q^{57} - q^{58} - 12 q^{59} - 10 q^{61} + 8 q^{62} - 4 q^{63} + 7 q^{64} - 4 q^{66} - 8 q^{67} + 6 q^{68} - 4 q^{69} - 8 q^{71} + 3 q^{72} + 2 q^{73} + 2 q^{74} + 4 q^{76} + 16 q^{77} - 6 q^{78} + q^{81} + 6 q^{82} - 8 q^{83} - 4 q^{84} + 4 q^{86} - q^{87} - 12 q^{88} - 6 q^{89} + 24 q^{91} - 4 q^{92} + 8 q^{93} + 5 q^{96} + 2 q^{97} - 9 q^{98} - 4 q^{99}+O(q^{100})$$ q - q^2 - q^3 - q^4 + q^6 - 4 * q^7 + 3 * q^8 + q^9 - 4 * q^11 + q^12 - 6 * q^13 + 4 * q^14 - q^16 - 6 * q^17 - q^18 - 4 * q^19 + 4 * q^21 + 4 * q^22 + 4 * q^23 - 3 * q^24 + 6 * q^26 - q^27 + 4 * q^28 + q^29 - 8 * q^31 - 5 * q^32 + 4 * q^33 + 6 * q^34 - q^36 - 2 * q^37 + 4 * q^38 + 6 * q^39 - 6 * q^41 - 4 * q^42 - 4 * q^43 + 4 * q^44 - 4 * q^46 + q^48 + 9 * q^49 + 6 * q^51 + 6 * q^52 + 10 * q^53 + q^54 - 12 * q^56 + 4 * q^57 - q^58 - 12 * q^59 - 10 * q^61 + 8 * q^62 - 4 * q^63 + 7 * q^64 - 4 * q^66 - 8 * q^67 + 6 * q^68 - 4 * q^69 - 8 * q^71 + 3 * q^72 + 2 * q^73 + 2 * q^74 + 4 * q^76 + 16 * q^77 - 6 * q^78 + q^81 + 6 * q^82 - 8 * q^83 - 4 * q^84 + 4 * q^86 - q^87 - 12 * q^88 - 6 * q^89 + 24 * q^91 - 4 * q^92 + 8 * q^93 + 5 * q^96 + 2 * q^97 - 9 * q^98 - 4 * 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
−1.00000 −1.00000 −1.00000 0 1.00000 −4.00000 3.00000 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.b 1
3.b odd 2 1 6525.2.a.j 1
5.b even 2 1 435.2.a.d 1
5.c odd 4 2 2175.2.c.b 2
15.d odd 2 1 1305.2.a.b 1
20.d odd 2 1 6960.2.a.l 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
435.2.a.d 1 5.b even 2 1
1305.2.a.b 1 15.d odd 2 1
2175.2.a.b 1 1.a even 1 1 trivial
2175.2.c.b 2 5.c odd 4 2
6525.2.a.j 1 3.b odd 2 1
6960.2.a.l 1 20.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2175))$$:

 $$T_{2} + 1$$ T2 + 1 $$T_{7} + 4$$ T7 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T + 1$$
$5$ $$T$$
$7$ $$T + 4$$
$11$ $$T + 4$$
$13$ $$T + 6$$
$17$ $$T + 6$$
$19$ $$T + 4$$
$23$ $$T - 4$$
$29$ $$T - 1$$
$31$ $$T + 8$$
$37$ $$T + 2$$
$41$ $$T + 6$$
$43$ $$T + 4$$
$47$ $$T$$
$53$ $$T - 10$$
$59$ $$T + 12$$
$61$ $$T + 10$$
$67$ $$T + 8$$
$71$ $$T + 8$$
$73$ $$T - 2$$
$79$ $$T$$
$83$ $$T + 8$$
$89$ $$T + 6$$
$97$ $$T - 2$$