# Properties

 Label 145.2.a.a Level $145$ Weight $2$ Character orbit 145.a Self dual yes Analytic conductor $1.158$ 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] = [145,2,Mod(1,145)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(145, 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("145.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$145 = 5 \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 145.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: $$1.15783082931$$ 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)

 $$f(q)$$ $$=$$ $$q - q^{2} - q^{4} - q^{5} - 2 q^{7} + 3 q^{8} - 3 q^{9}+O(q^{10})$$ q - q^2 - q^4 - q^5 - 2 * q^7 + 3 * q^8 - 3 * q^9 $$q - q^{2} - q^{4} - q^{5} - 2 q^{7} + 3 q^{8} - 3 q^{9} + q^{10} - 6 q^{11} + 2 q^{13} + 2 q^{14} - q^{16} - 2 q^{17} + 3 q^{18} - 2 q^{19} + q^{20} + 6 q^{22} + 2 q^{23} + q^{25} - 2 q^{26} + 2 q^{28} - q^{29} + 2 q^{31} - 5 q^{32} + 2 q^{34} + 2 q^{35} + 3 q^{36} + 10 q^{37} + 2 q^{38} - 3 q^{40} + 2 q^{41} + 8 q^{43} + 6 q^{44} + 3 q^{45} - 2 q^{46} - 12 q^{47} - 3 q^{49} - q^{50} - 2 q^{52} - 6 q^{53} + 6 q^{55} - 6 q^{56} + q^{58} - 8 q^{59} - 6 q^{61} - 2 q^{62} + 6 q^{63} + 7 q^{64} - 2 q^{65} + 2 q^{67} + 2 q^{68} - 2 q^{70} - 12 q^{71} - 9 q^{72} - 6 q^{73} - 10 q^{74} + 2 q^{76} + 12 q^{77} - 10 q^{79} + q^{80} + 9 q^{81} - 2 q^{82} - 14 q^{83} + 2 q^{85} - 8 q^{86} - 18 q^{88} + 18 q^{89} - 3 q^{90} - 4 q^{91} - 2 q^{92} + 12 q^{94} + 2 q^{95} + 2 q^{97} + 3 q^{98} + 18 q^{99}+O(q^{100})$$ q - q^2 - q^4 - q^5 - 2 * q^7 + 3 * q^8 - 3 * q^9 + q^10 - 6 * q^11 + 2 * q^13 + 2 * q^14 - q^16 - 2 * q^17 + 3 * q^18 - 2 * q^19 + q^20 + 6 * q^22 + 2 * q^23 + q^25 - 2 * q^26 + 2 * q^28 - q^29 + 2 * q^31 - 5 * q^32 + 2 * q^34 + 2 * q^35 + 3 * q^36 + 10 * q^37 + 2 * q^38 - 3 * q^40 + 2 * q^41 + 8 * q^43 + 6 * q^44 + 3 * q^45 - 2 * q^46 - 12 * q^47 - 3 * q^49 - q^50 - 2 * q^52 - 6 * q^53 + 6 * q^55 - 6 * q^56 + q^58 - 8 * q^59 - 6 * q^61 - 2 * q^62 + 6 * q^63 + 7 * q^64 - 2 * q^65 + 2 * q^67 + 2 * q^68 - 2 * q^70 - 12 * q^71 - 9 * q^72 - 6 * q^73 - 10 * q^74 + 2 * q^76 + 12 * q^77 - 10 * q^79 + q^80 + 9 * q^81 - 2 * q^82 - 14 * q^83 + 2 * q^85 - 8 * q^86 - 18 * q^88 + 18 * q^89 - 3 * q^90 - 4 * q^91 - 2 * q^92 + 12 * q^94 + 2 * q^95 + 2 * q^97 + 3 * q^98 + 18 * 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 0 −1.00000 −1.00000 0 −2.00000 3.00000 −3.00000 1.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

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

## 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 145.2.a.a 1
3.b odd 2 1 1305.2.a.f 1
4.b odd 2 1 2320.2.a.e 1
5.b even 2 1 725.2.a.a 1
5.c odd 4 2 725.2.b.a 2
7.b odd 2 1 7105.2.a.b 1
8.b even 2 1 9280.2.a.l 1
8.d odd 2 1 9280.2.a.o 1
15.d odd 2 1 6525.2.a.d 1
29.b even 2 1 4205.2.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
145.2.a.a 1 1.a even 1 1 trivial
725.2.a.a 1 5.b even 2 1
725.2.b.a 2 5.c odd 4 2
1305.2.a.f 1 3.b odd 2 1
2320.2.a.e 1 4.b odd 2 1
4205.2.a.a 1 29.b even 2 1
6525.2.a.d 1 15.d odd 2 1
7105.2.a.b 1 7.b odd 2 1
9280.2.a.l 1 8.b even 2 1
9280.2.a.o 1 8.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(145))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T$$
$5$ $$T + 1$$
$7$ $$T + 2$$
$11$ $$T + 6$$
$13$ $$T - 2$$
$17$ $$T + 2$$
$19$ $$T + 2$$
$23$ $$T - 2$$
$29$ $$T + 1$$
$31$ $$T - 2$$
$37$ $$T - 10$$
$41$ $$T - 2$$
$43$ $$T - 8$$
$47$ $$T + 12$$
$53$ $$T + 6$$
$59$ $$T + 8$$
$61$ $$T + 6$$
$67$ $$T - 2$$
$71$ $$T + 12$$
$73$ $$T + 6$$
$79$ $$T + 10$$
$83$ $$T + 14$$
$89$ $$T - 18$$
$97$ $$T - 2$$