# Properties

 Label 3450.2.a.bs Level $3450$ Weight $2$ Character orbit 3450.a Self dual yes Analytic conductor $27.548$ Analytic rank $0$ Dimension $3$ 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: $$3$$ Coefficient field: 3.3.3368.1 Defining polynomial: $$x^{3} - x^{2} - 15 x + 11$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{3} + q^{4} - q^{6} -\beta_{1} q^{7} + q^{8} + q^{9} +O(q^{10})$$ $$q + q^{2} - q^{3} + q^{4} - q^{6} -\beta_{1} q^{7} + q^{8} + q^{9} + ( 1 - \beta_{1} - \beta_{2} ) q^{11} - q^{12} + ( 1 - \beta_{1} + \beta_{2} ) q^{13} -\beta_{1} q^{14} + q^{16} + ( 1 + \beta_{2} ) q^{17} + q^{18} + ( -1 + \beta_{1} - \beta_{2} ) q^{19} + \beta_{1} q^{21} + ( 1 - \beta_{1} - \beta_{2} ) q^{22} + q^{23} - q^{24} + ( 1 - \beta_{1} + \beta_{2} ) q^{26} - q^{27} -\beta_{1} q^{28} - q^{29} + 6 q^{31} + q^{32} + ( -1 + \beta_{1} + \beta_{2} ) q^{33} + ( 1 + \beta_{2} ) q^{34} + q^{36} + ( -1 - \beta_{2} ) q^{37} + ( -1 + \beta_{1} - \beta_{2} ) q^{38} + ( -1 + \beta_{1} - \beta_{2} ) q^{39} + ( 7 - \beta_{1} + \beta_{2} ) q^{41} + \beta_{1} q^{42} + ( -5 + \beta_{1} - \beta_{2} ) q^{43} + ( 1 - \beta_{1} - \beta_{2} ) q^{44} + q^{46} + ( 2 + \beta_{1} - \beta_{2} ) q^{47} - q^{48} + ( 4 + 2 \beta_{2} ) q^{49} + ( -1 - \beta_{2} ) q^{51} + ( 1 - \beta_{1} + \beta_{2} ) q^{52} + ( -4 - 2 \beta_{2} ) q^{53} - q^{54} -\beta_{1} q^{56} + ( 1 - \beta_{1} + \beta_{2} ) q^{57} - q^{58} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{59} + ( 6 + 2 \beta_{1} ) q^{61} + 6 q^{62} -\beta_{1} q^{63} + q^{64} + ( -1 + \beta_{1} + \beta_{2} ) q^{66} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{67} + ( 1 + \beta_{2} ) q^{68} - q^{69} + ( 4 + \beta_{1} + \beta_{2} ) q^{71} + q^{72} + ( -1 + 2 \beta_{1} ) q^{73} + ( -1 - \beta_{2} ) q^{74} + ( -1 + \beta_{1} - \beta_{2} ) q^{76} + ( 11 + \beta_{1} + 3 \beta_{2} ) q^{77} + ( -1 + \beta_{1} - \beta_{2} ) q^{78} + ( -1 + \beta_{1} + \beta_{2} ) q^{79} + q^{81} + ( 7 - \beta_{1} + \beta_{2} ) q^{82} + ( -4 - 3 \beta_{1} ) q^{83} + \beta_{1} q^{84} + ( -5 + \beta_{1} - \beta_{2} ) q^{86} + q^{87} + ( 1 - \beta_{1} - \beta_{2} ) q^{88} + ( 3 + \beta_{2} ) q^{89} + ( 11 - 3 \beta_{1} + \beta_{2} ) q^{91} + q^{92} -6 q^{93} + ( 2 + \beta_{1} - \beta_{2} ) q^{94} - q^{96} + ( -4 + 2 \beta_{1} ) q^{97} + ( 4 + 2 \beta_{2} ) q^{98} + ( 1 - \beta_{1} - \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q + 3q^{2} - 3q^{3} + 3q^{4} - 3q^{6} - q^{7} + 3q^{8} + 3q^{9} + O(q^{10})$$ $$3q + 3q^{2} - 3q^{3} + 3q^{4} - 3q^{6} - q^{7} + 3q^{8} + 3q^{9} + 3q^{11} - 3q^{12} + q^{13} - q^{14} + 3q^{16} + 2q^{17} + 3q^{18} - q^{19} + q^{21} + 3q^{22} + 3q^{23} - 3q^{24} + q^{26} - 3q^{27} - q^{28} - 3q^{29} + 18q^{31} + 3q^{32} - 3q^{33} + 2q^{34} + 3q^{36} - 2q^{37} - q^{38} - q^{39} + 19q^{41} + q^{42} - 13q^{43} + 3q^{44} + 3q^{46} + 8q^{47} - 3q^{48} + 10q^{49} - 2q^{51} + q^{52} - 10q^{53} - 3q^{54} - q^{56} + q^{57} - 3q^{58} + 20q^{61} + 18q^{62} - q^{63} + 3q^{64} - 3q^{66} + 4q^{67} + 2q^{68} - 3q^{69} + 12q^{71} + 3q^{72} - q^{73} - 2q^{74} - q^{76} + 31q^{77} - q^{78} - 3q^{79} + 3q^{81} + 19q^{82} - 15q^{83} + q^{84} - 13q^{86} + 3q^{87} + 3q^{88} + 8q^{89} + 29q^{91} + 3q^{92} - 18q^{93} + 8q^{94} - 3q^{96} - 10q^{97} + 10q^{98} + 3q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 15 x + 11$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{2} - 11$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{2} + 11$$

## 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
 4.03932 0.723686 −3.76300
1.00000 −1.00000 1.00000 0 −1.00000 −4.03932 1.00000 1.00000 0
1.2 1.00000 −1.00000 1.00000 0 −1.00000 −0.723686 1.00000 1.00000 0
1.3 1.00000 −1.00000 1.00000 0 −1.00000 3.76300 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.bs yes 3
5.b even 2 1 3450.2.a.bp 3
5.c odd 4 2 3450.2.d.ba 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3450.2.a.bp 3 5.b even 2 1
3450.2.a.bs yes 3 1.a even 1 1 trivial
3450.2.d.ba 6 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3450))$$:

 $$T_{7}^{3} + T_{7}^{2} - 15 T_{7} - 11$$ $$T_{11}^{3} - 3 T_{11}^{2} - 32 T_{11} + 100$$ $$T_{13}^{3} - T_{13}^{2} - 32 T_{13} - 12$$ $$T_{17}^{3} - 2 T_{17}^{2} - 17 T_{17} + 40$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{3}$$
$3$ $$( 1 + T )^{3}$$
$5$ $$T^{3}$$
$7$ $$-11 - 15 T + T^{2} + T^{3}$$
$11$ $$100 - 32 T - 3 T^{2} + T^{3}$$
$13$ $$-12 - 32 T - T^{2} + T^{3}$$
$17$ $$40 - 17 T - 2 T^{2} + T^{3}$$
$19$ $$12 - 32 T + T^{2} + T^{3}$$
$23$ $$( -1 + T )^{3}$$
$29$ $$( 1 + T )^{3}$$
$31$ $$( -6 + T )^{3}$$
$37$ $$-40 - 17 T + 2 T^{2} + T^{3}$$
$41$ $$-72 + 88 T - 19 T^{2} + T^{3}$$
$43$ $$-36 + 24 T + 13 T^{2} + T^{3}$$
$47$ $$90 - 11 T - 8 T^{2} + T^{3}$$
$53$ $$-432 - 40 T + 10 T^{2} + T^{3}$$
$59$ $$-528 - 140 T + T^{3}$$
$61$ $$160 + 72 T - 20 T^{2} + T^{3}$$
$67$ $$352 - 124 T - 4 T^{2} + T^{3}$$
$71$ $$10 + 13 T - 12 T^{2} + T^{3}$$
$73$ $$27 - 61 T + T^{2} + T^{3}$$
$79$ $$-100 - 32 T + 3 T^{2} + T^{3}$$
$83$ $$-725 - 63 T + 15 T^{2} + T^{3}$$
$89$ $$58 + 3 T - 8 T^{2} + T^{3}$$
$97$ $$-120 - 28 T + 10 T^{2} + T^{3}$$