# Properties

 Label 2175.2.a.o Level $2175$ Weight $2$ Character orbit 2175.a Self dual yes Analytic conductor $17.367$ Analytic rank $1$ Dimension $2$ 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ 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)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{5}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta q^{2} - q^{3} + 3 q^{4} + \beta q^{6} - 2 q^{7} - \beta q^{8} + q^{9} +O(q^{10})$$ q - b * q^2 - q^3 + 3 * q^4 + b * q^6 - 2 * q^7 - b * q^8 + q^9 $$q - \beta q^{2} - q^{3} + 3 q^{4} + \beta q^{6} - 2 q^{7} - \beta q^{8} + q^{9} - 2 q^{11} - 3 q^{12} - 2 q^{13} + 2 \beta q^{14} - q^{16} + 2 \beta q^{17} - \beta q^{18} + 2 q^{19} + 2 q^{21} + 2 \beta q^{22} + 2 q^{23} + \beta q^{24} + 2 \beta q^{26} - q^{27} - 6 q^{28} - q^{29} - 2 q^{31} + 3 \beta q^{32} + 2 q^{33} - 10 q^{34} + 3 q^{36} + (2 \beta - 4) q^{37} - 2 \beta q^{38} + 2 q^{39} + 2 q^{41} - 2 \beta q^{42} - 4 q^{43} - 6 q^{44} - 2 \beta q^{46} + (4 \beta - 4) q^{47} + q^{48} - 3 q^{49} - 2 \beta q^{51} - 6 q^{52} - 2 q^{53} + \beta q^{54} + 2 \beta q^{56} - 2 q^{57} + \beta q^{58} + 8 q^{59} + ( - 4 \beta - 2) q^{61} + 2 \beta q^{62} - 2 q^{63} - 13 q^{64} - 2 \beta q^{66} + (4 \beta + 6) q^{67} + 6 \beta q^{68} - 2 q^{69} + 4 q^{71} - \beta q^{72} + ( - 2 \beta - 8) q^{73} + (4 \beta - 10) q^{74} + 6 q^{76} + 4 q^{77} - 2 \beta q^{78} + (4 \beta + 6) q^{79} + q^{81} - 2 \beta q^{82} + ( - 4 \beta + 6) q^{83} + 6 q^{84} + 4 \beta q^{86} + q^{87} + 2 \beta q^{88} - 6 q^{89} + 4 q^{91} + 6 q^{92} + 2 q^{93} + (4 \beta - 20) q^{94} - 3 \beta q^{96} + ( - 6 \beta + 4) q^{97} + 3 \beta q^{98} - 2 q^{99} +O(q^{100})$$ q - b * q^2 - q^3 + 3 * q^4 + b * q^6 - 2 * q^7 - b * q^8 + q^9 - 2 * q^11 - 3 * q^12 - 2 * q^13 + 2*b * q^14 - q^16 + 2*b * q^17 - b * q^18 + 2 * q^19 + 2 * q^21 + 2*b * q^22 + 2 * q^23 + b * q^24 + 2*b * q^26 - q^27 - 6 * q^28 - q^29 - 2 * q^31 + 3*b * q^32 + 2 * q^33 - 10 * q^34 + 3 * q^36 + (2*b - 4) * q^37 - 2*b * q^38 + 2 * q^39 + 2 * q^41 - 2*b * q^42 - 4 * q^43 - 6 * q^44 - 2*b * q^46 + (4*b - 4) * q^47 + q^48 - 3 * q^49 - 2*b * q^51 - 6 * q^52 - 2 * q^53 + b * q^54 + 2*b * q^56 - 2 * q^57 + b * q^58 + 8 * q^59 + (-4*b - 2) * q^61 + 2*b * q^62 - 2 * q^63 - 13 * q^64 - 2*b * q^66 + (4*b + 6) * q^67 + 6*b * q^68 - 2 * q^69 + 4 * q^71 - b * q^72 + (-2*b - 8) * q^73 + (4*b - 10) * q^74 + 6 * q^76 + 4 * q^77 - 2*b * q^78 + (4*b + 6) * q^79 + q^81 - 2*b * q^82 + (-4*b + 6) * q^83 + 6 * q^84 + 4*b * q^86 + q^87 + 2*b * q^88 - 6 * q^89 + 4 * q^91 + 6 * q^92 + 2 * q^93 + (4*b - 20) * q^94 - 3*b * q^96 + (-6*b + 4) * q^97 + 3*b * q^98 - 2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} + 6 q^{4} - 4 q^{7} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 + 6 * q^4 - 4 * q^7 + 2 * q^9 $$2 q - 2 q^{3} + 6 q^{4} - 4 q^{7} + 2 q^{9} - 4 q^{11} - 6 q^{12} - 4 q^{13} - 2 q^{16} + 4 q^{19} + 4 q^{21} + 4 q^{23} - 2 q^{27} - 12 q^{28} - 2 q^{29} - 4 q^{31} + 4 q^{33} - 20 q^{34} + 6 q^{36} - 8 q^{37} + 4 q^{39} + 4 q^{41} - 8 q^{43} - 12 q^{44} - 8 q^{47} + 2 q^{48} - 6 q^{49} - 12 q^{52} - 4 q^{53} - 4 q^{57} + 16 q^{59} - 4 q^{61} - 4 q^{63} - 26 q^{64} + 12 q^{67} - 4 q^{69} + 8 q^{71} - 16 q^{73} - 20 q^{74} + 12 q^{76} + 8 q^{77} + 12 q^{79} + 2 q^{81} + 12 q^{83} + 12 q^{84} + 2 q^{87} - 12 q^{89} + 8 q^{91} + 12 q^{92} + 4 q^{93} - 40 q^{94} + 8 q^{97} - 4 q^{99}+O(q^{100})$$ 2 * q - 2 * q^3 + 6 * q^4 - 4 * q^7 + 2 * q^9 - 4 * q^11 - 6 * q^12 - 4 * q^13 - 2 * q^16 + 4 * q^19 + 4 * q^21 + 4 * q^23 - 2 * q^27 - 12 * q^28 - 2 * q^29 - 4 * q^31 + 4 * q^33 - 20 * q^34 + 6 * q^36 - 8 * q^37 + 4 * q^39 + 4 * q^41 - 8 * q^43 - 12 * q^44 - 8 * q^47 + 2 * q^48 - 6 * q^49 - 12 * q^52 - 4 * q^53 - 4 * q^57 + 16 * q^59 - 4 * q^61 - 4 * q^63 - 26 * q^64 + 12 * q^67 - 4 * q^69 + 8 * q^71 - 16 * q^73 - 20 * q^74 + 12 * q^76 + 8 * q^77 + 12 * q^79 + 2 * q^81 + 12 * q^83 + 12 * q^84 + 2 * q^87 - 12 * q^89 + 8 * q^91 + 12 * q^92 + 4 * q^93 - 40 * q^94 + 8 * q^97 - 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
 1.61803 −0.618034
−2.23607 −1.00000 3.00000 0 2.23607 −2.00000 −2.23607 1.00000 0
1.2 2.23607 −1.00000 3.00000 0 −2.23607 −2.00000 2.23607 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.o 2
3.b odd 2 1 6525.2.a.y 2
5.b even 2 1 435.2.a.g 2
5.c odd 4 2 2175.2.c.g 4
15.d odd 2 1 1305.2.a.j 2
20.d odd 2 1 6960.2.a.bp 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
435.2.a.g 2 5.b even 2 1
1305.2.a.j 2 15.d odd 2 1
2175.2.a.o 2 1.a even 1 1 trivial
2175.2.c.g 4 5.c odd 4 2
6525.2.a.y 2 3.b odd 2 1
6960.2.a.bp 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2175))$$:

 $$T_{2}^{2} - 5$$ T2^2 - 5 $$T_{7} + 2$$ T7 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 5$$
$3$ $$(T + 1)^{2}$$
$5$ $$T^{2}$$
$7$ $$(T + 2)^{2}$$
$11$ $$(T + 2)^{2}$$
$13$ $$(T + 2)^{2}$$
$17$ $$T^{2} - 20$$
$19$ $$(T - 2)^{2}$$
$23$ $$(T - 2)^{2}$$
$29$ $$(T + 1)^{2}$$
$31$ $$(T + 2)^{2}$$
$37$ $$T^{2} + 8T - 4$$
$41$ $$(T - 2)^{2}$$
$43$ $$(T + 4)^{2}$$
$47$ $$T^{2} + 8T - 64$$
$53$ $$(T + 2)^{2}$$
$59$ $$(T - 8)^{2}$$
$61$ $$T^{2} + 4T - 76$$
$67$ $$T^{2} - 12T - 44$$
$71$ $$(T - 4)^{2}$$
$73$ $$T^{2} + 16T + 44$$
$79$ $$T^{2} - 12T - 44$$
$83$ $$T^{2} - 12T - 44$$
$89$ $$(T + 6)^{2}$$
$97$ $$T^{2} - 8T - 164$$