# Properties

 Label 3450.2.a.t Level $3450$ Weight $2$ Character orbit 3450.a Self dual yes Analytic conductor $27.548$ Analytic rank $1$ 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: $$1$$ 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

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{3} + q^{4} + q^{6} - 4 q^{7} + q^{8} + q^{9}+O(q^{10})$$ q + q^2 + q^3 + q^4 + q^6 - 4 * q^7 + q^8 + q^9 $$q + q^{2} + q^{3} + q^{4} + q^{6} - 4 q^{7} + q^{8} + q^{9} - 2 q^{11} + q^{12} - 4 q^{14} + q^{16} - 2 q^{17} + q^{18} - 4 q^{21} - 2 q^{22} - q^{23} + q^{24} + q^{27} - 4 q^{28} - 4 q^{29} + q^{32} - 2 q^{33} - 2 q^{34} + q^{36} - 10 q^{37} + 6 q^{41} - 4 q^{42} - 2 q^{43} - 2 q^{44} - q^{46} - 12 q^{47} + q^{48} + 9 q^{49} - 2 q^{51} - 6 q^{53} + q^{54} - 4 q^{56} - 4 q^{58} + 12 q^{59} - 14 q^{61} - 4 q^{63} + q^{64} - 2 q^{66} - 2 q^{67} - 2 q^{68} - q^{69} - 2 q^{71} + q^{72} - 6 q^{73} - 10 q^{74} + 8 q^{77} + 8 q^{79} + q^{81} + 6 q^{82} - 8 q^{83} - 4 q^{84} - 2 q^{86} - 4 q^{87} - 2 q^{88} - 8 q^{89} - q^{92} - 12 q^{94} + q^{96} + 9 q^{98} - 2 q^{99}+O(q^{100})$$ q + q^2 + q^3 + q^4 + q^6 - 4 * q^7 + q^8 + q^9 - 2 * q^11 + q^12 - 4 * q^14 + q^16 - 2 * q^17 + q^18 - 4 * q^21 - 2 * q^22 - q^23 + q^24 + q^27 - 4 * q^28 - 4 * q^29 + q^32 - 2 * q^33 - 2 * q^34 + q^36 - 10 * q^37 + 6 * q^41 - 4 * q^42 - 2 * q^43 - 2 * q^44 - q^46 - 12 * q^47 + q^48 + 9 * q^49 - 2 * q^51 - 6 * q^53 + q^54 - 4 * q^56 - 4 * q^58 + 12 * q^59 - 14 * q^61 - 4 * q^63 + q^64 - 2 * q^66 - 2 * q^67 - 2 * q^68 - q^69 - 2 * q^71 + q^72 - 6 * q^73 - 10 * q^74 + 8 * q^77 + 8 * q^79 + q^81 + 6 * q^82 - 8 * q^83 - 4 * q^84 - 2 * q^86 - 4 * q^87 - 2 * q^88 - 8 * q^89 - q^92 - 12 * q^94 + q^96 + 9 * q^98 - 2 * 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.

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 −4.00000 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.t 1
5.b even 2 1 690.2.a.b 1
5.c odd 4 2 3450.2.d.n 2
15.d odd 2 1 2070.2.a.s 1
20.d odd 2 1 5520.2.a.r 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
690.2.a.b 1 5.b even 2 1
2070.2.a.s 1 15.d odd 2 1
3450.2.a.t 1 1.a even 1 1 trivial
3450.2.d.n 2 5.c odd 4 2
5520.2.a.r 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3450))$$:

 $$T_{7} + 4$$ T7 + 4 $$T_{11} + 2$$ T11 + 2 $$T_{13}$$ T13 $$T_{17} + 2$$ T17 + 2

## Hecke characteristic polynomials

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