# Properties

 Label 1450.2.a.d Level $1450$ Weight $2$ Character orbit 1450.a Self dual yes Analytic conductor $11.578$ 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] = [1450,2,Mod(1,1450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1450, 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("1450.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1450 = 2 \cdot 5^{2} \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1450.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: $$11.5783082931$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ 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)

 $$f(q)$$ $$=$$ $$q - q^{2} + 2 q^{3} + q^{4} - 2 q^{6} - 2 q^{7} - q^{8} + q^{9}+O(q^{10})$$ q - q^2 + 2 * q^3 + q^4 - 2 * q^6 - 2 * q^7 - q^8 + q^9 $$q - q^{2} + 2 q^{3} + q^{4} - 2 q^{6} - 2 q^{7} - q^{8} + q^{9} - 6 q^{11} + 2 q^{12} - 2 q^{13} + 2 q^{14} + q^{16} + 6 q^{17} - q^{18} + 2 q^{19} - 4 q^{21} + 6 q^{22} - 2 q^{24} + 2 q^{26} - 4 q^{27} - 2 q^{28} - q^{29} - 7 q^{31} - q^{32} - 12 q^{33} - 6 q^{34} + q^{36} + 7 q^{37} - 2 q^{38} - 4 q^{39} - 12 q^{41} + 4 q^{42} - 8 q^{43} - 6 q^{44} - 9 q^{47} + 2 q^{48} - 3 q^{49} + 12 q^{51} - 2 q^{52} + 6 q^{53} + 4 q^{54} + 2 q^{56} + 4 q^{57} + q^{58} - 9 q^{59} - q^{61} + 7 q^{62} - 2 q^{63} + q^{64} + 12 q^{66} + q^{67} + 6 q^{68} - 12 q^{71} - q^{72} - 2 q^{73} - 7 q^{74} + 2 q^{76} + 12 q^{77} + 4 q^{78} + 8 q^{79} - 11 q^{81} + 12 q^{82} - 4 q^{84} + 8 q^{86} - 2 q^{87} + 6 q^{88} - 6 q^{89} + 4 q^{91} - 14 q^{93} + 9 q^{94} - 2 q^{96} + 4 q^{97} + 3 q^{98} - 6 q^{99}+O(q^{100})$$ q - q^2 + 2 * q^3 + q^4 - 2 * q^6 - 2 * q^7 - q^8 + q^9 - 6 * q^11 + 2 * q^12 - 2 * q^13 + 2 * q^14 + q^16 + 6 * q^17 - q^18 + 2 * q^19 - 4 * q^21 + 6 * q^22 - 2 * q^24 + 2 * q^26 - 4 * q^27 - 2 * q^28 - q^29 - 7 * q^31 - q^32 - 12 * q^33 - 6 * q^34 + q^36 + 7 * q^37 - 2 * q^38 - 4 * q^39 - 12 * q^41 + 4 * q^42 - 8 * q^43 - 6 * q^44 - 9 * q^47 + 2 * q^48 - 3 * q^49 + 12 * q^51 - 2 * q^52 + 6 * q^53 + 4 * q^54 + 2 * q^56 + 4 * q^57 + q^58 - 9 * q^59 - q^61 + 7 * q^62 - 2 * q^63 + q^64 + 12 * q^66 + q^67 + 6 * q^68 - 12 * q^71 - q^72 - 2 * q^73 - 7 * q^74 + 2 * q^76 + 12 * q^77 + 4 * q^78 + 8 * q^79 - 11 * q^81 + 12 * q^82 - 4 * q^84 + 8 * q^86 - 2 * q^87 + 6 * q^88 - 6 * q^89 + 4 * q^91 - 14 * q^93 + 9 * q^94 - 2 * q^96 + 4 * q^97 + 3 * 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
 0
−1.00000 2.00000 1.00000 0 −2.00000 −2.00000 −1.00000 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$5$$ $$1$$
$$29$$ $$1$$

## 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 1450.2.a.d 1
5.b even 2 1 1450.2.a.e yes 1
5.c odd 4 2 1450.2.b.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1450.2.a.d 1 1.a even 1 1 trivial
1450.2.a.e yes 1 5.b even 2 1
1450.2.b.a 2 5.c odd 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1450))$$:

 $$T_{3} - 2$$ T3 - 2 $$T_{7} + 2$$ T7 + 2 $$T_{13} + 2$$ T13 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T - 2$$
$5$ $$T$$
$7$ $$T + 2$$
$11$ $$T + 6$$
$13$ $$T + 2$$
$17$ $$T - 6$$
$19$ $$T - 2$$
$23$ $$T$$
$29$ $$T + 1$$
$31$ $$T + 7$$
$37$ $$T - 7$$
$41$ $$T + 12$$
$43$ $$T + 8$$
$47$ $$T + 9$$
$53$ $$T - 6$$
$59$ $$T + 9$$
$61$ $$T + 1$$
$67$ $$T - 1$$
$71$ $$T + 12$$
$73$ $$T + 2$$
$79$ $$T - 8$$
$83$ $$T$$
$89$ $$T + 6$$
$97$ $$T - 4$$