# Properties

 Label 5415.2.a.r Level $5415$ Weight $2$ Character orbit 5415.a Self dual yes Analytic conductor $43.239$ 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] = [5415,2,Mod(1,5415)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5415, 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("5415.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$5415 = 3 \cdot 5 \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5415.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: $$43.2389926945$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 285) 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 $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} - q^{3} + q^{4} + q^{5} - \beta q^{6} + ( - \beta - 1) q^{7} - \beta q^{8} + q^{9} +O(q^{10})$$ q + b * q^2 - q^3 + q^4 + q^5 - b * q^6 + (-b - 1) * q^7 - b * q^8 + q^9 $$q + \beta q^{2} - q^{3} + q^{4} + q^{5} - \beta q^{6} + ( - \beta - 1) q^{7} - \beta q^{8} + q^{9} + \beta q^{10} + (\beta + 3) q^{11} - q^{12} + ( - \beta + 1) q^{13} + ( - \beta - 3) q^{14} - q^{15} - 5 q^{16} + \beta q^{18} + q^{20} + (\beta + 1) q^{21} + (3 \beta + 3) q^{22} + 2 \beta q^{23} + \beta q^{24} + q^{25} + (\beta - 3) q^{26} - q^{27} + ( - \beta - 1) q^{28} + ( - 3 \beta - 3) q^{29} - \beta q^{30} + ( - 4 \beta - 2) q^{31} - 3 \beta q^{32} + ( - \beta - 3) q^{33} + ( - \beta - 1) q^{35} + q^{36} + (3 \beta + 1) q^{37} + (\beta - 1) q^{39} - \beta q^{40} + (\beta - 3) q^{41} + (\beta + 3) q^{42} + (3 \beta - 1) q^{43} + (\beta + 3) q^{44} + q^{45} + 6 q^{46} - 2 \beta q^{47} + 5 q^{48} + (2 \beta - 3) q^{49} + \beta q^{50} + ( - \beta + 1) q^{52} + (2 \beta + 6) q^{53} - \beta q^{54} + (\beta + 3) q^{55} + (\beta + 3) q^{56} + ( - 3 \beta - 9) q^{58} + (2 \beta - 6) q^{59} - q^{60} + (2 \beta - 10) q^{61} + ( - 2 \beta - 12) q^{62} + ( - \beta - 1) q^{63} + q^{64} + ( - \beta + 1) q^{65} + ( - 3 \beta - 3) q^{66} - 8 q^{67} - 2 \beta q^{69} + ( - \beta - 3) q^{70} + (6 \beta - 6) q^{71} - \beta q^{72} + ( - 4 \beta - 10) q^{73} + (\beta + 9) q^{74} - q^{75} + ( - 4 \beta - 6) q^{77} + ( - \beta + 3) q^{78} + (4 \beta + 4) q^{79} - 5 q^{80} + q^{81} + ( - 3 \beta + 3) q^{82} + ( - 4 \beta - 6) q^{83} + (\beta + 1) q^{84} + ( - \beta + 9) q^{86} + (3 \beta + 3) q^{87} + ( - 3 \beta - 3) q^{88} + ( - \beta - 9) q^{89} + \beta q^{90} + 2 q^{91} + 2 \beta q^{92} + (4 \beta + 2) q^{93} - 6 q^{94} + 3 \beta q^{96} + (3 \beta + 1) q^{97} + ( - 3 \beta + 6) q^{98} + (\beta + 3) q^{99} +O(q^{100})$$ q + b * q^2 - q^3 + q^4 + q^5 - b * q^6 + (-b - 1) * q^7 - b * q^8 + q^9 + b * q^10 + (b + 3) * q^11 - q^12 + (-b + 1) * q^13 + (-b - 3) * q^14 - q^15 - 5 * q^16 + b * q^18 + q^20 + (b + 1) * q^21 + (3*b + 3) * q^22 + 2*b * q^23 + b * q^24 + q^25 + (b - 3) * q^26 - q^27 + (-b - 1) * q^28 + (-3*b - 3) * q^29 - b * q^30 + (-4*b - 2) * q^31 - 3*b * q^32 + (-b - 3) * q^33 + (-b - 1) * q^35 + q^36 + (3*b + 1) * q^37 + (b - 1) * q^39 - b * q^40 + (b - 3) * q^41 + (b + 3) * q^42 + (3*b - 1) * q^43 + (b + 3) * q^44 + q^45 + 6 * q^46 - 2*b * q^47 + 5 * q^48 + (2*b - 3) * q^49 + b * q^50 + (-b + 1) * q^52 + (2*b + 6) * q^53 - b * q^54 + (b + 3) * q^55 + (b + 3) * q^56 + (-3*b - 9) * q^58 + (2*b - 6) * q^59 - q^60 + (2*b - 10) * q^61 + (-2*b - 12) * q^62 + (-b - 1) * q^63 + q^64 + (-b + 1) * q^65 + (-3*b - 3) * q^66 - 8 * q^67 - 2*b * q^69 + (-b - 3) * q^70 + (6*b - 6) * q^71 - b * q^72 + (-4*b - 10) * q^73 + (b + 9) * q^74 - q^75 + (-4*b - 6) * q^77 + (-b + 3) * q^78 + (4*b + 4) * q^79 - 5 * q^80 + q^81 + (-3*b + 3) * q^82 + (-4*b - 6) * q^83 + (b + 1) * q^84 + (-b + 9) * q^86 + (3*b + 3) * q^87 + (-3*b - 3) * q^88 + (-b - 9) * q^89 + b * q^90 + 2 * q^91 + 2*b * q^92 + (4*b + 2) * q^93 - 6 * q^94 + 3*b * q^96 + (3*b + 1) * q^97 + (-3*b + 6) * q^98 + (b + 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} + 2 q^{4} + 2 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 + 2 * q^4 + 2 * q^5 - 2 * q^7 + 2 * q^9 $$2 q - 2 q^{3} + 2 q^{4} + 2 q^{5} - 2 q^{7} + 2 q^{9} + 6 q^{11} - 2 q^{12} + 2 q^{13} - 6 q^{14} - 2 q^{15} - 10 q^{16} + 2 q^{20} + 2 q^{21} + 6 q^{22} + 2 q^{25} - 6 q^{26} - 2 q^{27} - 2 q^{28} - 6 q^{29} - 4 q^{31} - 6 q^{33} - 2 q^{35} + 2 q^{36} + 2 q^{37} - 2 q^{39} - 6 q^{41} + 6 q^{42} - 2 q^{43} + 6 q^{44} + 2 q^{45} + 12 q^{46} + 10 q^{48} - 6 q^{49} + 2 q^{52} + 12 q^{53} + 6 q^{55} + 6 q^{56} - 18 q^{58} - 12 q^{59} - 2 q^{60} - 20 q^{61} - 24 q^{62} - 2 q^{63} + 2 q^{64} + 2 q^{65} - 6 q^{66} - 16 q^{67} - 6 q^{70} - 12 q^{71} - 20 q^{73} + 18 q^{74} - 2 q^{75} - 12 q^{77} + 6 q^{78} + 8 q^{79} - 10 q^{80} + 2 q^{81} + 6 q^{82} - 12 q^{83} + 2 q^{84} + 18 q^{86} + 6 q^{87} - 6 q^{88} - 18 q^{89} + 4 q^{91} + 4 q^{93} - 12 q^{94} + 2 q^{97} + 12 q^{98} + 6 q^{99}+O(q^{100})$$ 2 * q - 2 * q^3 + 2 * q^4 + 2 * q^5 - 2 * q^7 + 2 * q^9 + 6 * q^11 - 2 * q^12 + 2 * q^13 - 6 * q^14 - 2 * q^15 - 10 * q^16 + 2 * q^20 + 2 * q^21 + 6 * q^22 + 2 * q^25 - 6 * q^26 - 2 * q^27 - 2 * q^28 - 6 * q^29 - 4 * q^31 - 6 * q^33 - 2 * q^35 + 2 * q^36 + 2 * q^37 - 2 * q^39 - 6 * q^41 + 6 * q^42 - 2 * q^43 + 6 * q^44 + 2 * q^45 + 12 * q^46 + 10 * q^48 - 6 * q^49 + 2 * q^52 + 12 * q^53 + 6 * q^55 + 6 * q^56 - 18 * q^58 - 12 * q^59 - 2 * q^60 - 20 * q^61 - 24 * q^62 - 2 * q^63 + 2 * q^64 + 2 * q^65 - 6 * q^66 - 16 * q^67 - 6 * q^70 - 12 * q^71 - 20 * q^73 + 18 * q^74 - 2 * q^75 - 12 * q^77 + 6 * q^78 + 8 * q^79 - 10 * q^80 + 2 * q^81 + 6 * q^82 - 12 * q^83 + 2 * q^84 + 18 * q^86 + 6 * q^87 - 6 * q^88 - 18 * q^89 + 4 * q^91 + 4 * q^93 - 12 * q^94 + 2 * q^97 + 12 * q^98 + 6 * 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
 −1.73205 1.73205
−1.73205 −1.00000 1.00000 1.00000 1.73205 0.732051 1.73205 1.00000 −1.73205
1.2 1.73205 −1.00000 1.00000 1.00000 −1.73205 −2.73205 −1.73205 1.00000 1.73205
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$
$$5$$ $$-1$$
$$19$$ $$-1$$

## 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 5415.2.a.r 2
19.b odd 2 1 285.2.a.e 2
57.d even 2 1 855.2.a.f 2
76.d even 2 1 4560.2.a.bh 2
95.d odd 2 1 1425.2.a.o 2
95.g even 4 2 1425.2.c.k 4
285.b even 2 1 4275.2.a.t 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
285.2.a.e 2 19.b odd 2 1
855.2.a.f 2 57.d even 2 1
1425.2.a.o 2 95.d odd 2 1
1425.2.c.k 4 95.g even 4 2
4275.2.a.t 2 285.b even 2 1
4560.2.a.bh 2 76.d even 2 1
5415.2.a.r 2 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(5415))$$:

 $$T_{2}^{2} - 3$$ T2^2 - 3 $$T_{7}^{2} + 2T_{7} - 2$$ T7^2 + 2*T7 - 2 $$T_{11}^{2} - 6T_{11} + 6$$ T11^2 - 6*T11 + 6 $$T_{13}^{2} - 2T_{13} - 2$$ T13^2 - 2*T13 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 3$$
$3$ $$(T + 1)^{2}$$
$5$ $$(T - 1)^{2}$$
$7$ $$T^{2} + 2T - 2$$
$11$ $$T^{2} - 6T + 6$$
$13$ $$T^{2} - 2T - 2$$
$17$ $$T^{2}$$
$19$ $$T^{2}$$
$23$ $$T^{2} - 12$$
$29$ $$T^{2} + 6T - 18$$
$31$ $$T^{2} + 4T - 44$$
$37$ $$T^{2} - 2T - 26$$
$41$ $$T^{2} + 6T + 6$$
$43$ $$T^{2} + 2T - 26$$
$47$ $$T^{2} - 12$$
$53$ $$T^{2} - 12T + 24$$
$59$ $$T^{2} + 12T + 24$$
$61$ $$T^{2} + 20T + 88$$
$67$ $$(T + 8)^{2}$$
$71$ $$T^{2} + 12T - 72$$
$73$ $$T^{2} + 20T + 52$$
$79$ $$T^{2} - 8T - 32$$
$83$ $$T^{2} + 12T - 12$$
$89$ $$T^{2} + 18T + 78$$
$97$ $$T^{2} - 2T - 26$$