# Properties

 Label 3450.2.a.z 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

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{3} + q^{4} + q^{6} + 2 q^{7} + q^{8} + q^{9}+O(q^{10})$$ q + q^2 + q^3 + q^4 + q^6 + 2 * q^7 + q^8 + q^9 $$q + q^{2} + q^{3} + q^{4} + q^{6} + 2 q^{7} + q^{8} + q^{9} + 2 q^{11} + q^{12} + 2 q^{13} + 2 q^{14} + q^{16} + q^{18} + 8 q^{19} + 2 q^{21} + 2 q^{22} + q^{23} + q^{24} + 2 q^{26} + q^{27} + 2 q^{28} - 10 q^{29} + 8 q^{31} + q^{32} + 2 q^{33} + q^{36} - 8 q^{37} + 8 q^{38} + 2 q^{39} - 6 q^{41} + 2 q^{42} - 12 q^{43} + 2 q^{44} + q^{46} - 8 q^{47} + q^{48} - 3 q^{49} + 2 q^{52} - 10 q^{53} + q^{54} + 2 q^{56} + 8 q^{57} - 10 q^{58} + 4 q^{59} + 12 q^{61} + 8 q^{62} + 2 q^{63} + q^{64} + 2 q^{66} + 4 q^{67} + q^{69} + 16 q^{71} + q^{72} + 10 q^{73} - 8 q^{74} + 8 q^{76} + 4 q^{77} + 2 q^{78} + 10 q^{79} + q^{81} - 6 q^{82} + 10 q^{83} + 2 q^{84} - 12 q^{86} - 10 q^{87} + 2 q^{88} + 4 q^{91} + q^{92} + 8 q^{93} - 8 q^{94} + q^{96} - 10 q^{97} - 3 q^{98} + 2 q^{99}+O(q^{100})$$ q + q^2 + q^3 + q^4 + q^6 + 2 * q^7 + q^8 + q^9 + 2 * q^11 + q^12 + 2 * q^13 + 2 * q^14 + q^16 + q^18 + 8 * q^19 + 2 * q^21 + 2 * q^22 + q^23 + q^24 + 2 * q^26 + q^27 + 2 * q^28 - 10 * q^29 + 8 * q^31 + q^32 + 2 * q^33 + q^36 - 8 * q^37 + 8 * q^38 + 2 * q^39 - 6 * q^41 + 2 * q^42 - 12 * q^43 + 2 * q^44 + q^46 - 8 * q^47 + q^48 - 3 * q^49 + 2 * q^52 - 10 * q^53 + q^54 + 2 * q^56 + 8 * q^57 - 10 * q^58 + 4 * q^59 + 12 * q^61 + 8 * q^62 + 2 * q^63 + q^64 + 2 * q^66 + 4 * q^67 + q^69 + 16 * q^71 + q^72 + 10 * q^73 - 8 * q^74 + 8 * q^76 + 4 * q^77 + 2 * q^78 + 10 * q^79 + q^81 - 6 * q^82 + 10 * q^83 + 2 * q^84 - 12 * q^86 - 10 * q^87 + 2 * q^88 + 4 * q^91 + q^92 + 8 * q^93 - 8 * q^94 + q^96 - 10 * q^97 - 3 * 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 2.00000 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.z 1
5.b even 2 1 690.2.a.c 1
5.c odd 4 2 3450.2.d.q 2
15.d odd 2 1 2070.2.a.k 1
20.d odd 2 1 5520.2.a.bg 1

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T - 1$$
$5$ $$T$$
$7$ $$T - 2$$
$11$ $$T - 2$$
$13$ $$T - 2$$
$17$ $$T$$
$19$ $$T - 8$$
$23$ $$T - 1$$
$29$ $$T + 10$$
$31$ $$T - 8$$
$37$ $$T + 8$$
$41$ $$T + 6$$
$43$ $$T + 12$$
$47$ $$T + 8$$
$53$ $$T + 10$$
$59$ $$T - 4$$
$61$ $$T - 12$$
$67$ $$T - 4$$
$71$ $$T - 16$$
$73$ $$T - 10$$
$79$ $$T - 10$$
$83$ $$T - 10$$
$89$ $$T$$
$97$ $$T + 10$$