# Properties

 Label 414.2.a.f Level $414$ Weight $2$ Character orbit 414.a Self dual yes Analytic conductor $3.306$ 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] = [414,2,Mod(1,414)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(414, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("414.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$414 = 2 \cdot 3^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 414.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: $$3.30580664368$$ 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 138) 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.

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} + (\beta + 1) q^{5} + 2 \beta q^{7} - q^{8}+O(q^{10})$$ q - q^2 + q^4 + (b + 1) * q^5 + 2*b * q^7 - q^8 $$q - q^{2} + q^{4} + (\beta + 1) q^{5} + 2 \beta q^{7} - q^{8} + ( - \beta - 1) q^{10} + ( - \beta + 3) q^{11} - 2 \beta q^{13} - 2 \beta q^{14} + q^{16} + 4 q^{17} + ( - 3 \beta - 1) q^{19} + (\beta + 1) q^{20} + (\beta - 3) q^{22} - q^{23} + (2 \beta + 1) q^{25} + 2 \beta q^{26} + 2 \beta q^{28} - 2 \beta q^{29} + (2 \beta + 2) q^{31} - q^{32} - 4 q^{34} + (2 \beta + 10) q^{35} + ( - \beta + 9) q^{37} + (3 \beta + 1) q^{38} + ( - \beta - 1) q^{40} + 2 q^{41} + ( - \beta - 7) q^{43} + ( - \beta + 3) q^{44} + q^{46} - 4 q^{47} + 13 q^{49} + ( - 2 \beta - 1) q^{50} - 2 \beta q^{52} + (\beta - 3) q^{53} + (2 \beta - 2) q^{55} - 2 \beta q^{56} + 2 \beta q^{58} - 4 \beta q^{59} + (\beta + 3) q^{61} + ( - 2 \beta - 2) q^{62} + q^{64} + ( - 2 \beta - 10) q^{65} + ( - 3 \beta + 3) q^{67} + 4 q^{68} + ( - 2 \beta - 10) q^{70} + 4 \beta q^{71} + 2 \beta q^{73} + (\beta - 9) q^{74} + ( - 3 \beta - 1) q^{76} + (6 \beta - 10) q^{77} - 2 \beta q^{79} + (\beta + 1) q^{80} - 2 q^{82} + (\beta - 11) q^{83} + (4 \beta + 4) q^{85} + (\beta + 7) q^{86} + (\beta - 3) q^{88} + ( - 2 \beta + 6) q^{89} - 20 q^{91} - q^{92} + 4 q^{94} + ( - 4 \beta - 16) q^{95} + ( - 2 \beta - 4) q^{97} - 13 q^{98} +O(q^{100})$$ q - q^2 + q^4 + (b + 1) * q^5 + 2*b * q^7 - q^8 + (-b - 1) * q^10 + (-b + 3) * q^11 - 2*b * q^13 - 2*b * q^14 + q^16 + 4 * q^17 + (-3*b - 1) * q^19 + (b + 1) * q^20 + (b - 3) * q^22 - q^23 + (2*b + 1) * q^25 + 2*b * q^26 + 2*b * q^28 - 2*b * q^29 + (2*b + 2) * q^31 - q^32 - 4 * q^34 + (2*b + 10) * q^35 + (-b + 9) * q^37 + (3*b + 1) * q^38 + (-b - 1) * q^40 + 2 * q^41 + (-b - 7) * q^43 + (-b + 3) * q^44 + q^46 - 4 * q^47 + 13 * q^49 + (-2*b - 1) * q^50 - 2*b * q^52 + (b - 3) * q^53 + (2*b - 2) * q^55 - 2*b * q^56 + 2*b * q^58 - 4*b * q^59 + (b + 3) * q^61 + (-2*b - 2) * q^62 + q^64 + (-2*b - 10) * q^65 + (-3*b + 3) * q^67 + 4 * q^68 + (-2*b - 10) * q^70 + 4*b * q^71 + 2*b * q^73 + (b - 9) * q^74 + (-3*b - 1) * q^76 + (6*b - 10) * q^77 - 2*b * q^79 + (b + 1) * q^80 - 2 * q^82 + (b - 11) * q^83 + (4*b + 4) * q^85 + (b + 7) * q^86 + (b - 3) * q^88 + (-2*b + 6) * q^89 - 20 * q^91 - q^92 + 4 * q^94 + (-4*b - 16) * q^95 + (-2*b - 4) * q^97 - 13 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - 2 q^{8}+O(q^{10})$$ 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^5 - 2 * q^8 $$2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - 2 q^{8} - 2 q^{10} + 6 q^{11} + 2 q^{16} + 8 q^{17} - 2 q^{19} + 2 q^{20} - 6 q^{22} - 2 q^{23} + 2 q^{25} + 4 q^{31} - 2 q^{32} - 8 q^{34} + 20 q^{35} + 18 q^{37} + 2 q^{38} - 2 q^{40} + 4 q^{41} - 14 q^{43} + 6 q^{44} + 2 q^{46} - 8 q^{47} + 26 q^{49} - 2 q^{50} - 6 q^{53} - 4 q^{55} + 6 q^{61} - 4 q^{62} + 2 q^{64} - 20 q^{65} + 6 q^{67} + 8 q^{68} - 20 q^{70} - 18 q^{74} - 2 q^{76} - 20 q^{77} + 2 q^{80} - 4 q^{82} - 22 q^{83} + 8 q^{85} + 14 q^{86} - 6 q^{88} + 12 q^{89} - 40 q^{91} - 2 q^{92} + 8 q^{94} - 32 q^{95} - 8 q^{97} - 26 q^{98}+O(q^{100})$$ 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^5 - 2 * q^8 - 2 * q^10 + 6 * q^11 + 2 * q^16 + 8 * q^17 - 2 * q^19 + 2 * q^20 - 6 * q^22 - 2 * q^23 + 2 * q^25 + 4 * q^31 - 2 * q^32 - 8 * q^34 + 20 * q^35 + 18 * q^37 + 2 * q^38 - 2 * q^40 + 4 * q^41 - 14 * q^43 + 6 * q^44 + 2 * q^46 - 8 * q^47 + 26 * q^49 - 2 * q^50 - 6 * q^53 - 4 * q^55 + 6 * q^61 - 4 * q^62 + 2 * q^64 - 20 * q^65 + 6 * q^67 + 8 * q^68 - 20 * q^70 - 18 * q^74 - 2 * q^76 - 20 * q^77 + 2 * q^80 - 4 * q^82 - 22 * q^83 + 8 * q^85 + 14 * q^86 - 6 * q^88 + 12 * q^89 - 40 * q^91 - 2 * q^92 + 8 * q^94 - 32 * q^95 - 8 * q^97 - 26 * 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
 −0.618034 1.61803
−1.00000 0 1.00000 −1.23607 0 −4.47214 −1.00000 0 1.23607
1.2 −1.00000 0 1.00000 3.23607 0 4.47214 −1.00000 0 −3.23607
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$-1$$
$$23$$ $$1$$

## 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 414.2.a.f 2
3.b odd 2 1 138.2.a.d 2
4.b odd 2 1 3312.2.a.bc 2
12.b even 2 1 1104.2.a.j 2
15.d odd 2 1 3450.2.a.be 2
15.e even 4 2 3450.2.d.x 4
21.c even 2 1 6762.2.a.cb 2
23.b odd 2 1 9522.2.a.q 2
24.f even 2 1 4416.2.a.bl 2
24.h odd 2 1 4416.2.a.bh 2
69.c even 2 1 3174.2.a.s 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
138.2.a.d 2 3.b odd 2 1
414.2.a.f 2 1.a even 1 1 trivial
1104.2.a.j 2 12.b even 2 1
3174.2.a.s 2 69.c even 2 1
3312.2.a.bc 2 4.b odd 2 1
3450.2.a.be 2 15.d odd 2 1
3450.2.d.x 4 15.e even 4 2
4416.2.a.bh 2 24.h odd 2 1
4416.2.a.bl 2 24.f even 2 1
6762.2.a.cb 2 21.c even 2 1
9522.2.a.q 2 23.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - 2T_{5} - 4$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(414))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 2T - 4$$
$7$ $$T^{2} - 20$$
$11$ $$T^{2} - 6T + 4$$
$13$ $$T^{2} - 20$$
$17$ $$(T - 4)^{2}$$
$19$ $$T^{2} + 2T - 44$$
$23$ $$(T + 1)^{2}$$
$29$ $$T^{2} - 20$$
$31$ $$T^{2} - 4T - 16$$
$37$ $$T^{2} - 18T + 76$$
$41$ $$(T - 2)^{2}$$
$43$ $$T^{2} + 14T + 44$$
$47$ $$(T + 4)^{2}$$
$53$ $$T^{2} + 6T + 4$$
$59$ $$T^{2} - 80$$
$61$ $$T^{2} - 6T + 4$$
$67$ $$T^{2} - 6T - 36$$
$71$ $$T^{2} - 80$$
$73$ $$T^{2} - 20$$
$79$ $$T^{2} - 20$$
$83$ $$T^{2} + 22T + 116$$
$89$ $$T^{2} - 12T + 16$$
$97$ $$T^{2} + 8T - 4$$