# Properties

 Label 3450.2.a.m 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: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{3} + q^{4} - q^{6} + 5q^{7} - q^{8} + q^{9} + O(q^{10})$$ $$q - q^{2} + q^{3} + q^{4} - q^{6} + 5q^{7} - q^{8} + q^{9} + q^{12} + 2q^{13} - 5q^{14} + q^{16} + 3q^{17} - q^{18} + 2q^{19} + 5q^{21} + q^{23} - q^{24} - 2q^{26} + q^{27} + 5q^{28} + 3q^{29} + 2q^{31} - q^{32} - 3q^{34} + q^{36} - 7q^{37} - 2q^{38} + 2q^{39} - 5q^{42} + 2q^{43} - q^{46} - 3q^{47} + q^{48} + 18q^{49} + 3q^{51} + 2q^{52} - 12q^{53} - q^{54} - 5q^{56} + 2q^{57} - 3q^{58} + 6q^{59} + 2q^{61} - 2q^{62} + 5q^{63} + q^{64} + 2q^{67} + 3q^{68} + q^{69} - 15q^{71} - q^{72} + 11q^{73} + 7q^{74} + 2q^{76} - 2q^{78} + 8q^{79} + q^{81} - 9q^{83} + 5q^{84} - 2q^{86} + 3q^{87} + 3q^{89} + 10q^{91} + q^{92} + 2q^{93} + 3q^{94} - q^{96} - 10q^{97} - 18q^{98} + O(q^{100})$$

## 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 1.00000 1.00000 0 −1.00000 5.00000 −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
$$2$$ $$1$$
$$3$$ $$-1$$
$$5$$ $$-1$$
$$23$$ $$-1$$

## 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.m 1
5.b even 2 1 3450.2.a.n yes 1
5.c odd 4 2 3450.2.d.h 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3450.2.a.m 1 1.a even 1 1 trivial
3450.2.a.n yes 1 5.b even 2 1
3450.2.d.h 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3450))$$:

 $$T_{7} - 5$$ $$T_{11}$$ $$T_{13} - 2$$ $$T_{17} - 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T$$
$3$ $$-1 + T$$
$5$ $$T$$
$7$ $$-5 + T$$
$11$ $$T$$
$13$ $$-2 + T$$
$17$ $$-3 + T$$
$19$ $$-2 + T$$
$23$ $$-1 + T$$
$29$ $$-3 + T$$
$31$ $$-2 + T$$
$37$ $$7 + T$$
$41$ $$T$$
$43$ $$-2 + T$$
$47$ $$3 + T$$
$53$ $$12 + T$$
$59$ $$-6 + T$$
$61$ $$-2 + T$$
$67$ $$-2 + T$$
$71$ $$15 + T$$
$73$ $$-11 + T$$
$79$ $$-8 + T$$
$83$ $$9 + T$$
$89$ $$-3 + T$$
$97$ $$10 + T$$