# Properties

 Label 3465.2.a.n Level $3465$ Weight $2$ Character orbit 3465.a Self dual yes Analytic conductor $27.668$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3465 = 3^{2} \cdot 5 \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3465.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$27.6681643004$$ Analytic rank: $$0$$ 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

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

## 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.

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.n 1
3.b odd 2 1 1155.2.a.f 1
15.d odd 2 1 5775.2.a.q 1
21.c even 2 1 8085.2.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1155.2.a.f 1 3.b odd 2 1
3465.2.a.n 1 1.a even 1 1 trivial
5775.2.a.q 1 15.d odd 2 1
8085.2.a.f 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3465))$$:

 $$T_{2} - 1$$ $$T_{13} + 2$$ $$T_{17} + 2$$ $$T_{19} - 4$$ $$T_{23}$$

## Hecke characteristic polynomials

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