# Properties

 Label 3450.2.d.s 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: no (minimal twist has level 690) 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.

 $$f(q)$$ $$=$$ $$q -i q^{2} + i q^{3} - q^{4} + q^{6} + 4 i q^{7} + i q^{8} - q^{9} +O(q^{10})$$ $$q -i q^{2} + i q^{3} - q^{4} + q^{6} + 4 i q^{7} + i q^{8} - q^{9} + 2 q^{11} -i q^{12} -4 i q^{13} + 4 q^{14} + q^{16} -6 i q^{17} + i q^{18} + 4 q^{19} -4 q^{21} -2 i q^{22} + i q^{23} - q^{24} -4 q^{26} -i q^{27} -4 i q^{28} -8 q^{29} + 8 q^{31} -i q^{32} + 2 i q^{33} -6 q^{34} + q^{36} -10 i q^{37} -4 i q^{38} + 4 q^{39} + 6 q^{41} + 4 i q^{42} -6 i q^{43} -2 q^{44} + q^{46} -4 i q^{47} + i q^{48} -9 q^{49} + 6 q^{51} + 4 i q^{52} + 14 i q^{53} - q^{54} -4 q^{56} + 4 i q^{57} + 8 i q^{58} -4 q^{59} + 6 q^{61} -8 i q^{62} -4 i q^{63} - q^{64} + 2 q^{66} + 14 i q^{67} + 6 i q^{68} - q^{69} + 10 q^{71} -i q^{72} -14 i q^{73} -10 q^{74} -4 q^{76} + 8 i q^{77} -4 i q^{78} + 8 q^{79} + q^{81} -6 i q^{82} + 4 i q^{83} + 4 q^{84} -6 q^{86} -8 i q^{87} + 2 i q^{88} + 16 q^{91} -i q^{92} + 8 i q^{93} -4 q^{94} + q^{96} -8 i q^{97} + 9 i q^{98} -2 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{4} + 2q^{6} - 2q^{9} + O(q^{10})$$ $$2q - 2q^{4} + 2q^{6} - 2q^{9} + 4q^{11} + 8q^{14} + 2q^{16} + 8q^{19} - 8q^{21} - 2q^{24} - 8q^{26} - 16q^{29} + 16q^{31} - 12q^{34} + 2q^{36} + 8q^{39} + 12q^{41} - 4q^{44} + 2q^{46} - 18q^{49} + 12q^{51} - 2q^{54} - 8q^{56} - 8q^{59} + 12q^{61} - 2q^{64} + 4q^{66} - 2q^{69} + 20q^{71} - 20q^{74} - 8q^{76} + 16q^{79} + 2q^{81} + 8q^{84} - 12q^{86} + 32q^{91} - 8q^{94} + 2q^{96} - 4q^{99} + O(q^{100})$$

## 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 4.00000i 1.00000i −1.00000 0
2899.2 1.00000i 1.00000i −1.00000 0 1.00000 4.00000i 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.s 2
5.b even 2 1 inner 3450.2.d.s 2
5.c odd 4 1 690.2.a.d 1
5.c odd 4 1 3450.2.a.u 1
15.e even 4 1 2070.2.a.o 1
20.e even 4 1 5520.2.a.bb 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
690.2.a.d 1 5.c odd 4 1
2070.2.a.o 1 15.e even 4 1
3450.2.a.u 1 5.c odd 4 1
3450.2.d.s 2 1.a even 1 1 trivial
3450.2.d.s 2 5.b even 2 1 inner
5520.2.a.bb 1 20.e even 4 1

## Hecke kernels

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

 $$T_{7}^{2} + 16$$ $$T_{11} - 2$$ $$T_{13}^{2} + 16$$ $$T_{17}^{2} + 36$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2}$$
$3$ $$1 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$16 + T^{2}$$
$11$ $$( -2 + T )^{2}$$
$13$ $$16 + T^{2}$$
$17$ $$36 + T^{2}$$
$19$ $$( -4 + T )^{2}$$
$23$ $$1 + T^{2}$$
$29$ $$( 8 + T )^{2}$$
$31$ $$( -8 + T )^{2}$$
$37$ $$100 + T^{2}$$
$41$ $$( -6 + T )^{2}$$
$43$ $$36 + T^{2}$$
$47$ $$16 + T^{2}$$
$53$ $$196 + T^{2}$$
$59$ $$( 4 + T )^{2}$$
$61$ $$( -6 + T )^{2}$$
$67$ $$196 + T^{2}$$
$71$ $$( -10 + T )^{2}$$
$73$ $$196 + T^{2}$$
$79$ $$( -8 + T )^{2}$$
$83$ $$16 + T^{2}$$
$89$ $$T^{2}$$
$97$ $$64 + T^{2}$$