# Properties

 Label 3465.2.a.g Level $3465$ Weight $2$ Character orbit 3465.a Self dual yes Analytic conductor $27.668$ 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] = [3465,2,Mod(1,3465)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3465, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("3465.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3465 = 3^{2} \cdot 5 \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3465.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: $$27.6681643004$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1155) 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)

 $$f(q)$$ $$=$$ $$q - q^{2} - q^{4} + q^{5} + q^{7} + 3 q^{8}+O(q^{10})$$ q - q^2 - q^4 + q^5 + q^7 + 3 * q^8 $$q - q^{2} - q^{4} + q^{5} + q^{7} + 3 q^{8} - q^{10} + q^{11} - 2 q^{13} - q^{14} - q^{16} - 6 q^{17} + 4 q^{19} - q^{20} - q^{22} + 4 q^{23} + q^{25} + 2 q^{26} - q^{28} - 10 q^{29} - 4 q^{31} - 5 q^{32} + 6 q^{34} + q^{35} - 2 q^{37} - 4 q^{38} + 3 q^{40} - 10 q^{41} + 12 q^{43} - q^{44} - 4 q^{46} + q^{49} - q^{50} + 2 q^{52} - 10 q^{53} + q^{55} + 3 q^{56} + 10 q^{58} + 12 q^{59} - 2 q^{61} + 4 q^{62} + 7 q^{64} - 2 q^{65} + 4 q^{67} + 6 q^{68} - q^{70} - 8 q^{71} - 14 q^{73} + 2 q^{74} - 4 q^{76} + q^{77} + 4 q^{79} - q^{80} + 10 q^{82} - 8 q^{83} - 6 q^{85} - 12 q^{86} + 3 q^{88} - 6 q^{89} - 2 q^{91} - 4 q^{92} + 4 q^{95} - 10 q^{97} - q^{98}+O(q^{100})$$ q - q^2 - q^4 + q^5 + q^7 + 3 * q^8 - q^10 + q^11 - 2 * q^13 - q^14 - q^16 - 6 * q^17 + 4 * q^19 - q^20 - q^22 + 4 * q^23 + q^25 + 2 * q^26 - q^28 - 10 * q^29 - 4 * q^31 - 5 * q^32 + 6 * q^34 + q^35 - 2 * q^37 - 4 * q^38 + 3 * q^40 - 10 * q^41 + 12 * q^43 - q^44 - 4 * q^46 + q^49 - q^50 + 2 * q^52 - 10 * q^53 + q^55 + 3 * q^56 + 10 * q^58 + 12 * q^59 - 2 * q^61 + 4 * q^62 + 7 * q^64 - 2 * q^65 + 4 * q^67 + 6 * q^68 - q^70 - 8 * q^71 - 14 * q^73 + 2 * q^74 - 4 * q^76 + q^77 + 4 * q^79 - q^80 + 10 * q^82 - 8 * q^83 - 6 * q^85 - 12 * q^86 + 3 * q^88 - 6 * q^89 - 2 * q^91 - 4 * q^92 + 4 * q^95 - 10 * q^97 - 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 1.00000 0 1.00000 3.00000 0 −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
$$3$$ $$-1$$
$$5$$ $$-1$$
$$7$$ $$-1$$
$$11$$ $$-1$$

## 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 3465.2.a.g 1
3.b odd 2 1 1155.2.a.j 1
15.d odd 2 1 5775.2.a.f 1
21.c even 2 1 8085.2.a.w 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1155.2.a.j 1 3.b odd 2 1
3465.2.a.g 1 1.a even 1 1 trivial
5775.2.a.f 1 15.d odd 2 1
8085.2.a.w 1 21.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3465))$$:

 $$T_{2} + 1$$ T2 + 1 $$T_{13} + 2$$ T13 + 2 $$T_{17} + 6$$ T17 + 6 $$T_{19} - 4$$ T19 - 4 $$T_{23} - 4$$ T23 - 4

## Hecke characteristic polynomials

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