# Properties

 Label 3450.2.d.k Level $3450$ Weight $2$ Character orbit 3450.d Analytic conductor $27.548$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3450.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$27.5483886973$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -i q^{2} + i q^{3} - q^{4} + q^{6} + i q^{7} + i q^{8} - q^{9} +O(q^{10})$$ $$q -i q^{2} + i q^{3} - q^{4} + q^{6} + i q^{7} + i q^{8} - q^{9} -5 q^{11} -i q^{12} -3 i q^{13} + q^{14} + q^{16} + 2 i q^{17} + i q^{18} + 3 q^{19} - q^{21} + 5 i q^{22} -i q^{23} - q^{24} -3 q^{26} -i q^{27} -i q^{28} + q^{29} -i q^{32} -5 i q^{33} + 2 q^{34} + q^{36} + 4 i q^{37} -3 i q^{38} + 3 q^{39} -3 q^{41} + i q^{42} -5 i q^{43} + 5 q^{44} - q^{46} -6 i q^{47} + i q^{48} + 6 q^{49} -2 q^{51} + 3 i q^{52} - q^{54} - q^{56} + 3 i q^{57} -i q^{58} + 6 q^{59} + 4 q^{61} -i q^{63} - q^{64} -5 q^{66} -4 i q^{67} -2 i q^{68} + q^{69} -2 q^{71} -i q^{72} -9 i q^{73} + 4 q^{74} -3 q^{76} -5 i q^{77} -3 i q^{78} + 7 q^{79} + q^{81} + 3 i q^{82} + i q^{83} + q^{84} -5 q^{86} + i q^{87} -5 i q^{88} + 2 q^{89} + 3 q^{91} + i q^{92} -6 q^{94} + q^{96} -6 i q^{98} + 5 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{4} + 2q^{6} - 2q^{9} + O(q^{10})$$ $$2q - 2q^{4} + 2q^{6} - 2q^{9} - 10q^{11} + 2q^{14} + 2q^{16} + 6q^{19} - 2q^{21} - 2q^{24} - 6q^{26} + 2q^{29} + 4q^{34} + 2q^{36} + 6q^{39} - 6q^{41} + 10q^{44} - 2q^{46} + 12q^{49} - 4q^{51} - 2q^{54} - 2q^{56} + 12q^{59} + 8q^{61} - 2q^{64} - 10q^{66} + 2q^{69} - 4q^{71} + 8q^{74} - 6q^{76} + 14q^{79} + 2q^{81} + 2q^{84} - 10q^{86} + 4q^{89} + 6q^{91} - 12q^{94} + 2q^{96} + 10q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3450\mathbb{Z}\right)^\times$$.

 $$n$$ $$277$$ $$1151$$ $$1201$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2899.1
 1.00000i − 1.00000i
1.00000i 1.00000i −1.00000 0 1.00000 1.00000i 1.00000i −1.00000 0
2899.2 1.00000i 1.00000i −1.00000 0 1.00000 1.00000i 1.00000i −1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3450.2.d.k 2
5.b even 2 1 inner 3450.2.d.k 2
5.c odd 4 1 3450.2.a.e 1
5.c odd 4 1 3450.2.a.w yes 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3450.2.a.e 1 5.c odd 4 1
3450.2.a.w yes 1 5.c odd 4 1
3450.2.d.k 2 1.a even 1 1 trivial
3450.2.d.k 2 5.b even 2 1 inner

## Hecke kernels

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

 $$T_{7}^{2} + 1$$ $$T_{11} + 5$$ $$T_{13}^{2} + 9$$ $$T_{17}^{2} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2}$$
$3$ $$1 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$1 + T^{2}$$
$11$ $$( 5 + T )^{2}$$
$13$ $$9 + T^{2}$$
$17$ $$4 + T^{2}$$
$19$ $$( -3 + T )^{2}$$
$23$ $$1 + T^{2}$$
$29$ $$( -1 + T )^{2}$$
$31$ $$T^{2}$$
$37$ $$16 + T^{2}$$
$41$ $$( 3 + T )^{2}$$
$43$ $$25 + T^{2}$$
$47$ $$36 + T^{2}$$
$53$ $$T^{2}$$
$59$ $$( -6 + T )^{2}$$
$61$ $$( -4 + T )^{2}$$
$67$ $$16 + T^{2}$$
$71$ $$( 2 + T )^{2}$$
$73$ $$81 + T^{2}$$
$79$ $$( -7 + T )^{2}$$
$83$ $$1 + T^{2}$$
$89$ $$( -2 + T )^{2}$$
$97$ $$T^{2}$$