# Properties

 Label 5415.2.a.u 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: $$\Q(\sqrt{2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 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 $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta + 1) q^{2} - q^{3} + (2 \beta + 1) q^{4} - q^{5} + ( - \beta - 1) q^{6} + ( - 2 \beta - 1) q^{7} + (\beta + 3) q^{8} + q^{9}+O(q^{10})$$ q + (b + 1) * q^2 - q^3 + (2*b + 1) * q^4 - q^5 + (-b - 1) * q^6 + (-2*b - 1) * q^7 + (b + 3) * q^8 + q^9 $$q + (\beta + 1) q^{2} - q^{3} + (2 \beta + 1) q^{4} - q^{5} + ( - \beta - 1) q^{6} + ( - 2 \beta - 1) q^{7} + (\beta + 3) q^{8} + q^{9} + ( - \beta - 1) q^{10} + 2 \beta q^{11} + ( - 2 \beta - 1) q^{12} + (2 \beta + 1) q^{13} + ( - 3 \beta - 5) q^{14} + q^{15} + 3 q^{16} + ( - 2 \beta - 4) q^{17} + (\beta + 1) q^{18} + ( - 2 \beta - 1) q^{20} + (2 \beta + 1) q^{21} + (2 \beta + 4) q^{22} + (2 \beta + 2) q^{23} + ( - \beta - 3) q^{24} + q^{25} + (3 \beta + 5) q^{26} - q^{27} + ( - 4 \beta - 9) q^{28} + ( - 4 \beta + 4) q^{29} + (\beta + 1) q^{30} - 5 q^{31} + (\beta - 3) q^{32} - 2 \beta q^{33} + ( - 6 \beta - 8) q^{34} + (2 \beta + 1) q^{35} + (2 \beta + 1) q^{36} + ( - 2 \beta - 5) q^{37} + ( - 2 \beta - 1) q^{39} + ( - \beta - 3) q^{40} - 2 \beta q^{41} + (3 \beta + 5) q^{42} + ( - 2 \beta + 5) q^{43} + (2 \beta + 8) q^{44} - q^{45} + (4 \beta + 6) q^{46} + ( - 2 \beta - 6) q^{47} - 3 q^{48} + (4 \beta + 2) q^{49} + (\beta + 1) q^{50} + (2 \beta + 4) q^{51} + (4 \beta + 9) q^{52} - 2 q^{53} + ( - \beta - 1) q^{54} - 2 \beta q^{55} + ( - 7 \beta - 7) q^{56} - 4 q^{58} + (2 \beta + 1) q^{60} + (8 \beta + 3) q^{61} + ( - 5 \beta - 5) q^{62} + ( - 2 \beta - 1) q^{63} + ( - 2 \beta - 7) q^{64} + ( - 2 \beta - 1) q^{65} + ( - 2 \beta - 4) q^{66} + ( - 6 \beta - 3) q^{67} + ( - 10 \beta - 12) q^{68} + ( - 2 \beta - 2) q^{69} + (3 \beta + 5) q^{70} + 10 q^{71} + (\beta + 3) q^{72} + ( - 6 \beta + 1) q^{73} + ( - 7 \beta - 9) q^{74} - q^{75} + ( - 2 \beta - 8) q^{77} + ( - 3 \beta - 5) q^{78} + ( - 4 \beta - 9) q^{79} - 3 q^{80} + q^{81} + ( - 2 \beta - 4) q^{82} - 8 q^{83} + (4 \beta + 9) q^{84} + (2 \beta + 4) q^{85} + (3 \beta + 1) q^{86} + (4 \beta - 4) q^{87} + (6 \beta + 4) q^{88} + ( - 6 \beta + 4) q^{89} + ( - \beta - 1) q^{90} + ( - 4 \beta - 9) q^{91} + (6 \beta + 10) q^{92} + 5 q^{93} + ( - 8 \beta - 10) q^{94} + ( - \beta + 3) q^{96} - 6 q^{97} + (6 \beta + 10) q^{98} + 2 \beta q^{99} +O(q^{100})$$ q + (b + 1) * q^2 - q^3 + (2*b + 1) * q^4 - q^5 + (-b - 1) * q^6 + (-2*b - 1) * q^7 + (b + 3) * q^8 + q^9 + (-b - 1) * q^10 + 2*b * q^11 + (-2*b - 1) * q^12 + (2*b + 1) * q^13 + (-3*b - 5) * q^14 + q^15 + 3 * q^16 + (-2*b - 4) * q^17 + (b + 1) * q^18 + (-2*b - 1) * q^20 + (2*b + 1) * q^21 + (2*b + 4) * q^22 + (2*b + 2) * q^23 + (-b - 3) * q^24 + q^25 + (3*b + 5) * q^26 - q^27 + (-4*b - 9) * q^28 + (-4*b + 4) * q^29 + (b + 1) * q^30 - 5 * q^31 + (b - 3) * q^32 - 2*b * q^33 + (-6*b - 8) * q^34 + (2*b + 1) * q^35 + (2*b + 1) * q^36 + (-2*b - 5) * q^37 + (-2*b - 1) * q^39 + (-b - 3) * q^40 - 2*b * q^41 + (3*b + 5) * q^42 + (-2*b + 5) * q^43 + (2*b + 8) * q^44 - q^45 + (4*b + 6) * q^46 + (-2*b - 6) * q^47 - 3 * q^48 + (4*b + 2) * q^49 + (b + 1) * q^50 + (2*b + 4) * q^51 + (4*b + 9) * q^52 - 2 * q^53 + (-b - 1) * q^54 - 2*b * q^55 + (-7*b - 7) * q^56 - 4 * q^58 + (2*b + 1) * q^60 + (8*b + 3) * q^61 + (-5*b - 5) * q^62 + (-2*b - 1) * q^63 + (-2*b - 7) * q^64 + (-2*b - 1) * q^65 + (-2*b - 4) * q^66 + (-6*b - 3) * q^67 + (-10*b - 12) * q^68 + (-2*b - 2) * q^69 + (3*b + 5) * q^70 + 10 * q^71 + (b + 3) * q^72 + (-6*b + 1) * q^73 + (-7*b - 9) * q^74 - q^75 + (-2*b - 8) * q^77 + (-3*b - 5) * q^78 + (-4*b - 9) * q^79 - 3 * q^80 + q^81 + (-2*b - 4) * q^82 - 8 * q^83 + (4*b + 9) * q^84 + (2*b + 4) * q^85 + (3*b + 1) * q^86 + (4*b - 4) * q^87 + (6*b + 4) * q^88 + (-6*b + 4) * q^89 + (-b - 1) * q^90 + (-4*b - 9) * q^91 + (6*b + 10) * q^92 + 5 * q^93 + (-8*b - 10) * q^94 + (-b + 3) * q^96 - 6 * q^97 + (6*b + 10) * q^98 + 2*b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} - 2 q^{7} + 6 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^5 - 2 * q^6 - 2 * q^7 + 6 * q^8 + 2 * q^9 $$2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} - 2 q^{7} + 6 q^{8} + 2 q^{9} - 2 q^{10} - 2 q^{12} + 2 q^{13} - 10 q^{14} + 2 q^{15} + 6 q^{16} - 8 q^{17} + 2 q^{18} - 2 q^{20} + 2 q^{21} + 8 q^{22} + 4 q^{23} - 6 q^{24} + 2 q^{25} + 10 q^{26} - 2 q^{27} - 18 q^{28} + 8 q^{29} + 2 q^{30} - 10 q^{31} - 6 q^{32} - 16 q^{34} + 2 q^{35} + 2 q^{36} - 10 q^{37} - 2 q^{39} - 6 q^{40} + 10 q^{42} + 10 q^{43} + 16 q^{44} - 2 q^{45} + 12 q^{46} - 12 q^{47} - 6 q^{48} + 4 q^{49} + 2 q^{50} + 8 q^{51} + 18 q^{52} - 4 q^{53} - 2 q^{54} - 14 q^{56} - 8 q^{58} + 2 q^{60} + 6 q^{61} - 10 q^{62} - 2 q^{63} - 14 q^{64} - 2 q^{65} - 8 q^{66} - 6 q^{67} - 24 q^{68} - 4 q^{69} + 10 q^{70} + 20 q^{71} + 6 q^{72} + 2 q^{73} - 18 q^{74} - 2 q^{75} - 16 q^{77} - 10 q^{78} - 18 q^{79} - 6 q^{80} + 2 q^{81} - 8 q^{82} - 16 q^{83} + 18 q^{84} + 8 q^{85} + 2 q^{86} - 8 q^{87} + 8 q^{88} + 8 q^{89} - 2 q^{90} - 18 q^{91} + 20 q^{92} + 10 q^{93} - 20 q^{94} + 6 q^{96} - 12 q^{97} + 20 q^{98}+O(q^{100})$$ 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^5 - 2 * q^6 - 2 * q^7 + 6 * q^8 + 2 * q^9 - 2 * q^10 - 2 * q^12 + 2 * q^13 - 10 * q^14 + 2 * q^15 + 6 * q^16 - 8 * q^17 + 2 * q^18 - 2 * q^20 + 2 * q^21 + 8 * q^22 + 4 * q^23 - 6 * q^24 + 2 * q^25 + 10 * q^26 - 2 * q^27 - 18 * q^28 + 8 * q^29 + 2 * q^30 - 10 * q^31 - 6 * q^32 - 16 * q^34 + 2 * q^35 + 2 * q^36 - 10 * q^37 - 2 * q^39 - 6 * q^40 + 10 * q^42 + 10 * q^43 + 16 * q^44 - 2 * q^45 + 12 * q^46 - 12 * q^47 - 6 * q^48 + 4 * q^49 + 2 * q^50 + 8 * q^51 + 18 * q^52 - 4 * q^53 - 2 * q^54 - 14 * q^56 - 8 * q^58 + 2 * q^60 + 6 * q^61 - 10 * q^62 - 2 * q^63 - 14 * q^64 - 2 * q^65 - 8 * q^66 - 6 * q^67 - 24 * q^68 - 4 * q^69 + 10 * q^70 + 20 * q^71 + 6 * q^72 + 2 * q^73 - 18 * q^74 - 2 * q^75 - 16 * q^77 - 10 * q^78 - 18 * q^79 - 6 * q^80 + 2 * q^81 - 8 * q^82 - 16 * q^83 + 18 * q^84 + 8 * q^85 + 2 * q^86 - 8 * q^87 + 8 * q^88 + 8 * q^89 - 2 * q^90 - 18 * q^91 + 20 * q^92 + 10 * q^93 - 20 * q^94 + 6 * q^96 - 12 * q^97 + 20 * q^98

## 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
 −1.41421 1.41421
−0.414214 −1.00000 −1.82843 −1.00000 0.414214 1.82843 1.58579 1.00000 0.414214
1.2 2.41421 −1.00000 3.82843 −1.00000 −2.41421 −3.82843 4.41421 1.00000 −2.41421
 $$n$$: 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.u 2
19.b odd 2 1 5415.2.a.o 2
19.c even 3 2 285.2.i.d 4
57.h odd 6 2 855.2.k.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
285.2.i.d 4 19.c even 3 2
855.2.k.f 4 57.h odd 6 2
5415.2.a.o 2 19.b odd 2 1
5415.2.a.u 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(5415))$$:

 $$T_{2}^{2} - 2T_{2} - 1$$ T2^2 - 2*T2 - 1 $$T_{7}^{2} + 2T_{7} - 7$$ T7^2 + 2*T7 - 7 $$T_{11}^{2} - 8$$ T11^2 - 8 $$T_{13}^{2} - 2T_{13} - 7$$ T13^2 - 2*T13 - 7

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 2T - 1$$
$3$ $$(T + 1)^{2}$$
$5$ $$(T + 1)^{2}$$
$7$ $$T^{2} + 2T - 7$$
$11$ $$T^{2} - 8$$
$13$ $$T^{2} - 2T - 7$$
$17$ $$T^{2} + 8T + 8$$
$19$ $$T^{2}$$
$23$ $$T^{2} - 4T - 4$$
$29$ $$T^{2} - 8T - 16$$
$31$ $$(T + 5)^{2}$$
$37$ $$T^{2} + 10T + 17$$
$41$ $$T^{2} - 8$$
$43$ $$T^{2} - 10T + 17$$
$47$ $$T^{2} + 12T + 28$$
$53$ $$(T + 2)^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2} - 6T - 119$$
$67$ $$T^{2} + 6T - 63$$
$71$ $$(T - 10)^{2}$$
$73$ $$T^{2} - 2T - 71$$
$79$ $$T^{2} + 18T + 49$$
$83$ $$(T + 8)^{2}$$
$89$ $$T^{2} - 8T - 56$$
$97$ $$(T + 6)^{2}$$