Properties

 Label 414.2.a.b Level $414$ Weight $2$ Character orbit 414.a Self dual yes Analytic conductor $3.306$ Analytic rank $1$ Dimension $1$ 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: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 46) 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)

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} - 4 q^{5} - 4 q^{7} + q^{8}+O(q^{10})$$ q + q^2 + q^4 - 4 * q^5 - 4 * q^7 + q^8 $$q + q^{2} + q^{4} - 4 q^{5} - 4 q^{7} + q^{8} - 4 q^{10} - 2 q^{11} - 2 q^{13} - 4 q^{14} + q^{16} + 2 q^{17} - 2 q^{19} - 4 q^{20} - 2 q^{22} - q^{23} + 11 q^{25} - 2 q^{26} - 4 q^{28} - 2 q^{29} + q^{32} + 2 q^{34} + 16 q^{35} - 4 q^{37} - 2 q^{38} - 4 q^{40} - 6 q^{41} + 10 q^{43} - 2 q^{44} - q^{46} + 9 q^{49} + 11 q^{50} - 2 q^{52} + 4 q^{53} + 8 q^{55} - 4 q^{56} - 2 q^{58} - 12 q^{59} - 8 q^{61} + q^{64} + 8 q^{65} - 10 q^{67} + 2 q^{68} + 16 q^{70} + 6 q^{73} - 4 q^{74} - 2 q^{76} + 8 q^{77} - 12 q^{79} - 4 q^{80} - 6 q^{82} - 14 q^{83} - 8 q^{85} + 10 q^{86} - 2 q^{88} + 6 q^{89} + 8 q^{91} - q^{92} + 8 q^{95} + 6 q^{97} + 9 q^{98}+O(q^{100})$$ q + q^2 + q^4 - 4 * q^5 - 4 * q^7 + q^8 - 4 * q^10 - 2 * q^11 - 2 * q^13 - 4 * q^14 + q^16 + 2 * q^17 - 2 * q^19 - 4 * q^20 - 2 * q^22 - q^23 + 11 * q^25 - 2 * q^26 - 4 * q^28 - 2 * q^29 + q^32 + 2 * q^34 + 16 * q^35 - 4 * q^37 - 2 * q^38 - 4 * q^40 - 6 * q^41 + 10 * q^43 - 2 * q^44 - q^46 + 9 * q^49 + 11 * q^50 - 2 * q^52 + 4 * q^53 + 8 * q^55 - 4 * q^56 - 2 * q^58 - 12 * q^59 - 8 * q^61 + q^64 + 8 * q^65 - 10 * q^67 + 2 * q^68 + 16 * q^70 + 6 * q^73 - 4 * q^74 - 2 * q^76 + 8 * q^77 - 12 * q^79 - 4 * q^80 - 6 * q^82 - 14 * q^83 - 8 * q^85 + 10 * q^86 - 2 * q^88 + 6 * q^89 + 8 * q^91 - q^92 + 8 * q^95 + 6 * q^97 + 9 * 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
1.00000 0 1.00000 −4.00000 0 −4.00000 1.00000 0 −4.00000
 $$n$$: 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.b 1
3.b odd 2 1 46.2.a.a 1
4.b odd 2 1 3312.2.a.b 1
12.b even 2 1 368.2.a.d 1
15.d odd 2 1 1150.2.a.h 1
15.e even 4 2 1150.2.b.d 2
21.c even 2 1 2254.2.a.c 1
23.b odd 2 1 9522.2.a.p 1
24.f even 2 1 1472.2.a.g 1
24.h odd 2 1 1472.2.a.f 1
33.d even 2 1 5566.2.a.h 1
39.d odd 2 1 7774.2.a.d 1
60.h even 2 1 9200.2.a.p 1
69.c even 2 1 1058.2.a.c 1
276.h odd 2 1 8464.2.a.g 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
46.2.a.a 1 3.b odd 2 1
368.2.a.d 1 12.b even 2 1
414.2.a.b 1 1.a even 1 1 trivial
1058.2.a.c 1 69.c even 2 1
1150.2.a.h 1 15.d odd 2 1
1150.2.b.d 2 15.e even 4 2
1472.2.a.f 1 24.h odd 2 1
1472.2.a.g 1 24.f even 2 1
2254.2.a.c 1 21.c even 2 1
3312.2.a.b 1 4.b odd 2 1
5566.2.a.h 1 33.d even 2 1
7774.2.a.d 1 39.d odd 2 1
8464.2.a.g 1 276.h odd 2 1
9200.2.a.p 1 60.h even 2 1
9522.2.a.p 1 23.b odd 2 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T$$
$5$ $$T + 4$$
$7$ $$T + 4$$
$11$ $$T + 2$$
$13$ $$T + 2$$
$17$ $$T - 2$$
$19$ $$T + 2$$
$23$ $$T + 1$$
$29$ $$T + 2$$
$31$ $$T$$
$37$ $$T + 4$$
$41$ $$T + 6$$
$43$ $$T - 10$$
$47$ $$T$$
$53$ $$T - 4$$
$59$ $$T + 12$$
$61$ $$T + 8$$
$67$ $$T + 10$$
$71$ $$T$$
$73$ $$T - 6$$
$79$ $$T + 12$$
$83$ $$T + 14$$
$89$ $$T - 6$$
$97$ $$T - 6$$