# Properties

 Label 414.2.a.g 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: $$\Q(\sqrt{7})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 7$$ x^2 - 7 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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 $$\beta = \sqrt{7}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} + (\beta + 1) q^{5} + 2 q^{7} + q^{8}+O(q^{10})$$ q + q^2 + q^4 + (b + 1) * q^5 + 2 * q^7 + q^8 $$q + q^{2} + q^{4} + (\beta + 1) q^{5} + 2 q^{7} + q^{8} + (\beta + 1) q^{10} + ( - \beta - 1) q^{11} - 2 \beta q^{13} + 2 q^{14} + q^{16} + ( - 2 \beta - 2) q^{17} + (\beta + 3) q^{19} + (\beta + 1) q^{20} + ( - \beta - 1) q^{22} - q^{23} + (2 \beta + 3) q^{25} - 2 \beta q^{26} + 2 q^{28} + (2 \beta - 4) q^{29} + (2 \beta + 4) q^{31} + q^{32} + ( - 2 \beta - 2) q^{34} + (2 \beta + 2) q^{35} + ( - 3 \beta - 1) q^{37} + (\beta + 3) q^{38} + (\beta + 1) q^{40} - 6 q^{41} + (\beta + 3) q^{43} + ( - \beta - 1) q^{44} - q^{46} - 6 q^{47} - 3 q^{49} + (2 \beta + 3) q^{50} - 2 \beta q^{52} + ( - \beta - 1) q^{53} + ( - 2 \beta - 8) q^{55} + 2 q^{56} + (2 \beta - 4) q^{58} + (\beta + 3) q^{61} + (2 \beta + 4) q^{62} + q^{64} + ( - 2 \beta - 14) q^{65} + (3 \beta - 7) q^{67} + ( - 2 \beta - 2) q^{68} + (2 \beta + 2) q^{70} + 6 q^{71} + (2 \beta - 2) q^{73} + ( - 3 \beta - 1) q^{74} + (\beta + 3) q^{76} + ( - 2 \beta - 2) q^{77} + ( - 4 \beta - 2) q^{79} + (\beta + 1) q^{80} - 6 q^{82} + ( - \beta + 11) q^{83} + ( - 4 \beta - 16) q^{85} + (\beta + 3) q^{86} + ( - \beta - 1) q^{88} - 4 \beta q^{91} - q^{92} - 6 q^{94} + (4 \beta + 10) q^{95} + (2 \beta + 4) q^{97} - 3 q^{98} +O(q^{100})$$ q + q^2 + q^4 + (b + 1) * q^5 + 2 * q^7 + q^8 + (b + 1) * q^10 + (-b - 1) * q^11 - 2*b * q^13 + 2 * q^14 + q^16 + (-2*b - 2) * q^17 + (b + 3) * q^19 + (b + 1) * q^20 + (-b - 1) * q^22 - q^23 + (2*b + 3) * q^25 - 2*b * q^26 + 2 * q^28 + (2*b - 4) * q^29 + (2*b + 4) * q^31 + q^32 + (-2*b - 2) * q^34 + (2*b + 2) * q^35 + (-3*b - 1) * q^37 + (b + 3) * q^38 + (b + 1) * q^40 - 6 * q^41 + (b + 3) * q^43 + (-b - 1) * q^44 - q^46 - 6 * q^47 - 3 * q^49 + (2*b + 3) * q^50 - 2*b * q^52 + (-b - 1) * q^53 + (-2*b - 8) * q^55 + 2 * q^56 + (2*b - 4) * q^58 + (b + 3) * q^61 + (2*b + 4) * q^62 + q^64 + (-2*b - 14) * q^65 + (3*b - 7) * q^67 + (-2*b - 2) * q^68 + (2*b + 2) * q^70 + 6 * q^71 + (2*b - 2) * q^73 + (-3*b - 1) * q^74 + (b + 3) * q^76 + (-2*b - 2) * q^77 + (-4*b - 2) * q^79 + (b + 1) * q^80 - 6 * q^82 + (-b + 11) * q^83 + (-4*b - 16) * q^85 + (b + 3) * q^86 + (-b - 1) * q^88 - 4*b * q^91 - q^92 - 6 * q^94 + (4*b + 10) * q^95 + (2*b + 4) * q^97 - 3 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + 4 q^{7} + 2 q^{8}+O(q^{10})$$ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^5 + 4 * q^7 + 2 * q^8 $$2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + 4 q^{7} + 2 q^{8} + 2 q^{10} - 2 q^{11} + 4 q^{14} + 2 q^{16} - 4 q^{17} + 6 q^{19} + 2 q^{20} - 2 q^{22} - 2 q^{23} + 6 q^{25} + 4 q^{28} - 8 q^{29} + 8 q^{31} + 2 q^{32} - 4 q^{34} + 4 q^{35} - 2 q^{37} + 6 q^{38} + 2 q^{40} - 12 q^{41} + 6 q^{43} - 2 q^{44} - 2 q^{46} - 12 q^{47} - 6 q^{49} + 6 q^{50} - 2 q^{53} - 16 q^{55} + 4 q^{56} - 8 q^{58} + 6 q^{61} + 8 q^{62} + 2 q^{64} - 28 q^{65} - 14 q^{67} - 4 q^{68} + 4 q^{70} + 12 q^{71} - 4 q^{73} - 2 q^{74} + 6 q^{76} - 4 q^{77} - 4 q^{79} + 2 q^{80} - 12 q^{82} + 22 q^{83} - 32 q^{85} + 6 q^{86} - 2 q^{88} - 2 q^{92} - 12 q^{94} + 20 q^{95} + 8 q^{97} - 6 q^{98}+O(q^{100})$$ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^5 + 4 * q^7 + 2 * q^8 + 2 * q^10 - 2 * q^11 + 4 * q^14 + 2 * q^16 - 4 * q^17 + 6 * q^19 + 2 * q^20 - 2 * q^22 - 2 * q^23 + 6 * q^25 + 4 * q^28 - 8 * q^29 + 8 * q^31 + 2 * q^32 - 4 * q^34 + 4 * q^35 - 2 * q^37 + 6 * q^38 + 2 * q^40 - 12 * q^41 + 6 * q^43 - 2 * q^44 - 2 * q^46 - 12 * q^47 - 6 * q^49 + 6 * q^50 - 2 * q^53 - 16 * q^55 + 4 * q^56 - 8 * q^58 + 6 * q^61 + 8 * q^62 + 2 * q^64 - 28 * q^65 - 14 * q^67 - 4 * q^68 + 4 * q^70 + 12 * q^71 - 4 * q^73 - 2 * q^74 + 6 * q^76 - 4 * q^77 - 4 * q^79 + 2 * q^80 - 12 * q^82 + 22 * q^83 - 32 * q^85 + 6 * q^86 - 2 * q^88 - 2 * q^92 - 12 * q^94 + 20 * q^95 + 8 * q^97 - 6 * 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
 −2.64575 2.64575
1.00000 0 1.00000 −1.64575 0 2.00000 1.00000 0 −1.64575
1.2 1.00000 0 1.00000 3.64575 0 2.00000 1.00000 0 3.64575
 $$n$$: 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.g yes 2
3.b odd 2 1 414.2.a.e 2
4.b odd 2 1 3312.2.a.z 2
12.b even 2 1 3312.2.a.v 2
23.b odd 2 1 9522.2.a.bc 2
69.c even 2 1 9522.2.a.bb 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
414.2.a.e 2 3.b odd 2 1
414.2.a.g yes 2 1.a even 1 1 trivial
3312.2.a.v 2 12.b even 2 1
3312.2.a.z 2 4.b odd 2 1
9522.2.a.bb 2 69.c even 2 1
9522.2.a.bc 2 23.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - 2T_{5} - 6$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(414))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 2T - 6$$
$7$ $$(T - 2)^{2}$$
$11$ $$T^{2} + 2T - 6$$
$13$ $$T^{2} - 28$$
$17$ $$T^{2} + 4T - 24$$
$19$ $$T^{2} - 6T + 2$$
$23$ $$(T + 1)^{2}$$
$29$ $$T^{2} + 8T - 12$$
$31$ $$T^{2} - 8T - 12$$
$37$ $$T^{2} + 2T - 62$$
$41$ $$(T + 6)^{2}$$
$43$ $$T^{2} - 6T + 2$$
$47$ $$(T + 6)^{2}$$
$53$ $$T^{2} + 2T - 6$$
$59$ $$T^{2}$$
$61$ $$T^{2} - 6T + 2$$
$67$ $$T^{2} + 14T - 14$$
$71$ $$(T - 6)^{2}$$
$73$ $$T^{2} + 4T - 24$$
$79$ $$T^{2} + 4T - 108$$
$83$ $$T^{2} - 22T + 114$$
$89$ $$T^{2}$$
$97$ $$T^{2} - 8T - 12$$