# Properties

 Label 3450.2.a.d Level $3450$ Weight $2$ Character orbit 3450.a Self dual yes Analytic conductor $27.548$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3450.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$27.5483886973$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 690) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{2} - q^{3} + q^{4} + q^{6} - q^{8} + q^{9} + O(q^{10})$$ $$q - q^{2} - q^{3} + q^{4} + q^{6} - q^{8} + q^{9} + 4q^{11} - q^{12} + 2q^{13} + q^{16} + 6q^{17} - q^{18} + 4q^{19} - 4q^{22} + q^{23} + q^{24} - 2q^{26} - q^{27} - 2q^{29} - q^{32} - 4q^{33} - 6q^{34} + q^{36} + 2q^{37} - 4q^{38} - 2q^{39} + 10q^{41} + 4q^{43} + 4q^{44} - q^{46} - q^{48} - 7q^{49} - 6q^{51} + 2q^{52} - 6q^{53} + q^{54} - 4q^{57} + 2q^{58} - 4q^{59} - 10q^{61} + q^{64} + 4q^{66} + 12q^{67} + 6q^{68} - q^{69} - 8q^{71} - q^{72} - 10q^{73} - 2q^{74} + 4q^{76} + 2q^{78} - 8q^{79} + q^{81} - 10q^{82} + 4q^{83} - 4q^{86} + 2q^{87} - 4q^{88} + 18q^{89} + q^{92} + q^{96} - 2q^{97} + 7q^{98} + 4q^{99} + O(q^{100})$$

## 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 −1.00000 1.00000 0 1.00000 0 −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
$$2$$ $$1$$
$$3$$ $$1$$
$$5$$ $$1$$
$$23$$ $$-1$$

## 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 3450.2.a.d 1
5.b even 2 1 690.2.a.k 1
5.c odd 4 2 3450.2.d.t 2
15.d odd 2 1 2070.2.a.b 1
20.d odd 2 1 5520.2.a.i 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
690.2.a.k 1 5.b even 2 1
2070.2.a.b 1 15.d odd 2 1
3450.2.a.d 1 1.a even 1 1 trivial
3450.2.d.t 2 5.c odd 4 2
5520.2.a.i 1 20.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3450))$$:

 $$T_{7}$$ $$T_{11} - 4$$ $$T_{13} - 2$$ $$T_{17} - 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T$$
$3$ $$1 + T$$
$5$ $$T$$
$7$ $$T$$
$11$ $$-4 + T$$
$13$ $$-2 + T$$
$17$ $$-6 + T$$
$19$ $$-4 + T$$
$23$ $$-1 + T$$
$29$ $$2 + T$$
$31$ $$T$$
$37$ $$-2 + T$$
$41$ $$-10 + T$$
$43$ $$-4 + T$$
$47$ $$T$$
$53$ $$6 + T$$
$59$ $$4 + T$$
$61$ $$10 + T$$
$67$ $$-12 + T$$
$71$ $$8 + T$$
$73$ $$10 + T$$
$79$ $$8 + T$$
$83$ $$-4 + T$$
$89$ $$-18 + T$$
$97$ $$2 + T$$