# Properties

 Label 2175.2.a.q Level $2175$ Weight $2$ Character orbit 2175.a Self dual yes Analytic conductor $17.367$ Analytic rank $0$ 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: $$0$$ 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: $$1$$ 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)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{5})$$. We also show the integral $$q$$-expansion of the trace form.

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

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

 $$T_{2}^{2} - T_{2} - 1$$ T2^2 - T2 - 1 $$T_{7} - 3$$ T7 - 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T - 1$$
$3$ $$(T + 1)^{2}$$
$5$ $$T^{2}$$
$7$ $$(T - 3)^{2}$$
$11$ $$T^{2} + 2T - 19$$
$13$ $$T^{2} - 8T + 11$$
$17$ $$T^{2} - 2T - 19$$
$19$ $$T^{2} - 2T - 44$$
$23$ $$T^{2}$$
$29$ $$(T - 1)^{2}$$
$31$ $$(T + 8)^{2}$$
$37$ $$(T - 8)^{2}$$
$41$ $$T^{2} - 20$$
$43$ $$T^{2} + 2T - 4$$
$47$ $$T^{2} - 45$$
$53$ $$T^{2} - 18T + 76$$
$59$ $$T^{2} - 6T + 4$$
$61$ $$T^{2} - 2T - 44$$
$67$ $$T^{2} - 14T + 29$$
$71$ $$T^{2} - 10T + 20$$
$73$ $$(T - 8)^{2}$$
$79$ $$T^{2} - 10T - 100$$
$83$ $$T^{2} + 6T - 36$$
$89$ $$T^{2} - 125$$
$97$ $$T^{2} + 10T + 20$$