# Properties

 Label 2175.2.a.r 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: $$\Q(\sqrt{21})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 5$$ x^2 - x - 5 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 $$\beta = \frac{1}{2}(1 + \sqrt{21})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} - q^{3} + (\beta + 3) q^{4} - \beta q^{6} - q^{7} + (2 \beta + 5) q^{8} + q^{9} +O(q^{10})$$ q + b * q^2 - q^3 + (b + 3) * q^4 - b * q^6 - q^7 + (2*b + 5) * q^8 + q^9 $$q + \beta q^{2} - q^{3} + (\beta + 3) q^{4} - \beta q^{6} - q^{7} + (2 \beta + 5) q^{8} + q^{9} + 5 q^{11} + ( - \beta - 3) q^{12} + ( - 2 \beta + 1) q^{13} - \beta q^{14} + (5 \beta + 4) q^{16} + 3 q^{17} + \beta q^{18} - 2 \beta q^{19} + q^{21} + 5 \beta q^{22} + 4 q^{23} + ( - 2 \beta - 5) q^{24} + ( - \beta - 10) q^{26} - q^{27} + ( - \beta - 3) q^{28} + q^{29} + 4 q^{31} + (5 \beta + 15) q^{32} - 5 q^{33} + 3 \beta q^{34} + (\beta + 3) q^{36} + 4 q^{37} + ( - 2 \beta - 10) q^{38} + (2 \beta - 1) q^{39} + (4 \beta - 2) q^{41} + \beta q^{42} + ( - 2 \beta + 6) q^{43} + (5 \beta + 15) q^{44} + 4 \beta q^{46} + (2 \beta - 7) q^{47} + ( - 5 \beta - 4) q^{48} - 6 q^{49} - 3 q^{51} + ( - 7 \beta - 7) q^{52} + ( - 2 \beta - 4) q^{53} - \beta q^{54} + ( - 2 \beta - 5) q^{56} + 2 \beta q^{57} + \beta q^{58} + (2 \beta - 4) q^{59} + ( - 6 \beta + 2) q^{61} + 4 \beta q^{62} - q^{63} + (10 \beta + 17) q^{64} - 5 \beta q^{66} + ( - 4 \beta - 3) q^{67} + (3 \beta + 9) q^{68} - 4 q^{69} + (2 \beta - 6) q^{71} + (2 \beta + 5) q^{72} - 4 q^{73} + 4 \beta q^{74} + ( - 8 \beta - 10) q^{76} - 5 q^{77} + (\beta + 10) q^{78} + ( - 2 \beta + 4) q^{79} + q^{81} + (2 \beta + 20) q^{82} + ( - 2 \beta + 8) q^{83} + (\beta + 3) q^{84} + (4 \beta - 10) q^{86} - q^{87} + (10 \beta + 25) q^{88} + (2 \beta + 5) q^{89} + (2 \beta - 1) q^{91} + (4 \beta + 12) q^{92} - 4 q^{93} + ( - 5 \beta + 10) q^{94} + ( - 5 \beta - 15) q^{96} + (2 \beta - 8) q^{97} - 6 \beta q^{98} + 5 q^{99} +O(q^{100})$$ q + b * q^2 - q^3 + (b + 3) * q^4 - b * q^6 - q^7 + (2*b + 5) * q^8 + q^9 + 5 * q^11 + (-b - 3) * q^12 + (-2*b + 1) * q^13 - b * q^14 + (5*b + 4) * q^16 + 3 * q^17 + b * q^18 - 2*b * q^19 + q^21 + 5*b * q^22 + 4 * q^23 + (-2*b - 5) * q^24 + (-b - 10) * q^26 - q^27 + (-b - 3) * q^28 + q^29 + 4 * q^31 + (5*b + 15) * q^32 - 5 * q^33 + 3*b * q^34 + (b + 3) * q^36 + 4 * q^37 + (-2*b - 10) * q^38 + (2*b - 1) * q^39 + (4*b - 2) * q^41 + b * q^42 + (-2*b + 6) * q^43 + (5*b + 15) * q^44 + 4*b * q^46 + (2*b - 7) * q^47 + (-5*b - 4) * q^48 - 6 * q^49 - 3 * q^51 + (-7*b - 7) * q^52 + (-2*b - 4) * q^53 - b * q^54 + (-2*b - 5) * q^56 + 2*b * q^57 + b * q^58 + (2*b - 4) * q^59 + (-6*b + 2) * q^61 + 4*b * q^62 - q^63 + (10*b + 17) * q^64 - 5*b * q^66 + (-4*b - 3) * q^67 + (3*b + 9) * q^68 - 4 * q^69 + (2*b - 6) * q^71 + (2*b + 5) * q^72 - 4 * q^73 + 4*b * q^74 + (-8*b - 10) * q^76 - 5 * q^77 + (b + 10) * q^78 + (-2*b + 4) * q^79 + q^81 + (2*b + 20) * q^82 + (-2*b + 8) * q^83 + (b + 3) * q^84 + (4*b - 10) * q^86 - q^87 + (10*b + 25) * q^88 + (2*b + 5) * q^89 + (2*b - 1) * q^91 + (4*b + 12) * q^92 - 4 * q^93 + (-5*b + 10) * q^94 + (-5*b - 15) * q^96 + (2*b - 8) * q^97 - 6*b * q^98 + 5 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} - 2 q^{3} + 7 q^{4} - q^{6} - 2 q^{7} + 12 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q + q^2 - 2 * q^3 + 7 * q^4 - q^6 - 2 * q^7 + 12 * q^8 + 2 * q^9 $$2 q + q^{2} - 2 q^{3} + 7 q^{4} - q^{6} - 2 q^{7} + 12 q^{8} + 2 q^{9} + 10 q^{11} - 7 q^{12} - q^{14} + 13 q^{16} + 6 q^{17} + q^{18} - 2 q^{19} + 2 q^{21} + 5 q^{22} + 8 q^{23} - 12 q^{24} - 21 q^{26} - 2 q^{27} - 7 q^{28} + 2 q^{29} + 8 q^{31} + 35 q^{32} - 10 q^{33} + 3 q^{34} + 7 q^{36} + 8 q^{37} - 22 q^{38} + q^{42} + 10 q^{43} + 35 q^{44} + 4 q^{46} - 12 q^{47} - 13 q^{48} - 12 q^{49} - 6 q^{51} - 21 q^{52} - 10 q^{53} - q^{54} - 12 q^{56} + 2 q^{57} + q^{58} - 6 q^{59} - 2 q^{61} + 4 q^{62} - 2 q^{63} + 44 q^{64} - 5 q^{66} - 10 q^{67} + 21 q^{68} - 8 q^{69} - 10 q^{71} + 12 q^{72} - 8 q^{73} + 4 q^{74} - 28 q^{76} - 10 q^{77} + 21 q^{78} + 6 q^{79} + 2 q^{81} + 42 q^{82} + 14 q^{83} + 7 q^{84} - 16 q^{86} - 2 q^{87} + 60 q^{88} + 12 q^{89} + 28 q^{92} - 8 q^{93} + 15 q^{94} - 35 q^{96} - 14 q^{97} - 6 q^{98} + 10 q^{99}+O(q^{100})$$ 2 * q + q^2 - 2 * q^3 + 7 * q^4 - q^6 - 2 * q^7 + 12 * q^8 + 2 * q^9 + 10 * q^11 - 7 * q^12 - q^14 + 13 * q^16 + 6 * q^17 + q^18 - 2 * q^19 + 2 * q^21 + 5 * q^22 + 8 * q^23 - 12 * q^24 - 21 * q^26 - 2 * q^27 - 7 * q^28 + 2 * q^29 + 8 * q^31 + 35 * q^32 - 10 * q^33 + 3 * q^34 + 7 * q^36 + 8 * q^37 - 22 * q^38 + q^42 + 10 * q^43 + 35 * q^44 + 4 * q^46 - 12 * q^47 - 13 * q^48 - 12 * q^49 - 6 * q^51 - 21 * q^52 - 10 * q^53 - q^54 - 12 * q^56 + 2 * q^57 + q^58 - 6 * q^59 - 2 * q^61 + 4 * q^62 - 2 * q^63 + 44 * q^64 - 5 * q^66 - 10 * q^67 + 21 * q^68 - 8 * q^69 - 10 * q^71 + 12 * q^72 - 8 * q^73 + 4 * q^74 - 28 * q^76 - 10 * q^77 + 21 * q^78 + 6 * q^79 + 2 * q^81 + 42 * q^82 + 14 * q^83 + 7 * q^84 - 16 * q^86 - 2 * q^87 + 60 * q^88 + 12 * q^89 + 28 * q^92 - 8 * q^93 + 15 * q^94 - 35 * q^96 - 14 * q^97 - 6 * q^98 + 10 * 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
 −1.79129 2.79129
−1.79129 −1.00000 1.20871 0 1.79129 −1.00000 1.41742 1.00000 0
1.2 2.79129 −1.00000 5.79129 0 −2.79129 −1.00000 10.5826 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.r 2
3.b odd 2 1 6525.2.a.t 2
5.b even 2 1 435.2.a.f 2
5.c odd 4 2 2175.2.c.f 4
15.d odd 2 1 1305.2.a.m 2
20.d odd 2 1 6960.2.a.bw 2

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

 $$T_{2}^{2} - T_{2} - 5$$ T2^2 - T2 - 5 $$T_{7} + 1$$ T7 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T - 5$$
$3$ $$(T + 1)^{2}$$
$5$ $$T^{2}$$
$7$ $$(T + 1)^{2}$$
$11$ $$(T - 5)^{2}$$
$13$ $$T^{2} - 21$$
$17$ $$(T - 3)^{2}$$
$19$ $$T^{2} + 2T - 20$$
$23$ $$(T - 4)^{2}$$
$29$ $$(T - 1)^{2}$$
$31$ $$(T - 4)^{2}$$
$37$ $$(T - 4)^{2}$$
$41$ $$T^{2} - 84$$
$43$ $$T^{2} - 10T + 4$$
$47$ $$T^{2} + 12T + 15$$
$53$ $$T^{2} + 10T + 4$$
$59$ $$T^{2} + 6T - 12$$
$61$ $$T^{2} + 2T - 188$$
$67$ $$T^{2} + 10T - 59$$
$71$ $$T^{2} + 10T + 4$$
$73$ $$(T + 4)^{2}$$
$79$ $$T^{2} - 6T - 12$$
$83$ $$T^{2} - 14T + 28$$
$89$ $$T^{2} - 12T + 15$$
$97$ $$T^{2} + 14T + 28$$