# Properties

 Label 345.2.a.i Level $345$ Weight $2$ Character orbit 345.a Self dual yes Analytic conductor $2.755$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$345 = 3 \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 345.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$2.75483886973$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{6})$$ Defining polynomial: $$x^{2} - 6$$ Coefficient ring: $$\Z[a_1, a_2]$$ 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 $$\beta = \sqrt{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + q^{3} + 4 q^{4} - q^{5} + \beta q^{6} - q^{7} + 2 \beta q^{8} + q^{9} +O(q^{10})$$ $$q + \beta q^{2} + q^{3} + 4 q^{4} - q^{5} + \beta q^{6} - q^{7} + 2 \beta q^{8} + q^{9} -\beta q^{10} -\beta q^{11} + 4 q^{12} + ( 2 - \beta ) q^{13} -\beta q^{14} - q^{15} + 4 q^{16} + ( -3 + \beta ) q^{17} + \beta q^{18} + ( 2 - \beta ) q^{19} -4 q^{20} - q^{21} -6 q^{22} - q^{23} + 2 \beta q^{24} + q^{25} + ( -6 + 2 \beta ) q^{26} + q^{27} -4 q^{28} + ( 3 - 3 \beta ) q^{29} -\beta q^{30} + ( 5 + 2 \beta ) q^{31} -\beta q^{33} + ( 6 - 3 \beta ) q^{34} + q^{35} + 4 q^{36} + ( -1 - 2 \beta ) q^{37} + ( -6 + 2 \beta ) q^{38} + ( 2 - \beta ) q^{39} -2 \beta q^{40} + ( 3 - \beta ) q^{41} -\beta q^{42} + 2 q^{43} -4 \beta q^{44} - q^{45} -\beta q^{46} + ( 6 - \beta ) q^{47} + 4 q^{48} -6 q^{49} + \beta q^{50} + ( -3 + \beta ) q^{51} + ( 8 - 4 \beta ) q^{52} + ( 3 + \beta ) q^{53} + \beta q^{54} + \beta q^{55} -2 \beta q^{56} + ( 2 - \beta ) q^{57} + ( -18 + 3 \beta ) q^{58} + ( 3 - 3 \beta ) q^{59} -4 q^{60} + ( 8 + 3 \beta ) q^{61} + ( 12 + 5 \beta ) q^{62} - q^{63} -8 q^{64} + ( -2 + \beta ) q^{65} -6 q^{66} -7 q^{67} + ( -12 + 4 \beta ) q^{68} - q^{69} + \beta q^{70} + ( 3 + 3 \beta ) q^{71} + 2 \beta q^{72} + ( 2 + 3 \beta ) q^{73} + ( -12 - \beta ) q^{74} + q^{75} + ( 8 - 4 \beta ) q^{76} + \beta q^{77} + ( -6 + 2 \beta ) q^{78} -4 q^{79} -4 q^{80} + q^{81} + ( -6 + 3 \beta ) q^{82} + ( 3 - 5 \beta ) q^{83} -4 q^{84} + ( 3 - \beta ) q^{85} + 2 \beta q^{86} + ( 3 - 3 \beta ) q^{87} -12 q^{88} + ( -12 + 2 \beta ) q^{89} -\beta q^{90} + ( -2 + \beta ) q^{91} -4 q^{92} + ( 5 + 2 \beta ) q^{93} + ( -6 + 6 \beta ) q^{94} + ( -2 + \beta ) q^{95} + ( 8 + 2 \beta ) q^{97} -6 \beta q^{98} -\beta q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{3} + 8q^{4} - 2q^{5} - 2q^{7} + 2q^{9} + O(q^{10})$$ $$2q + 2q^{3} + 8q^{4} - 2q^{5} - 2q^{7} + 2q^{9} + 8q^{12} + 4q^{13} - 2q^{15} + 8q^{16} - 6q^{17} + 4q^{19} - 8q^{20} - 2q^{21} - 12q^{22} - 2q^{23} + 2q^{25} - 12q^{26} + 2q^{27} - 8q^{28} + 6q^{29} + 10q^{31} + 12q^{34} + 2q^{35} + 8q^{36} - 2q^{37} - 12q^{38} + 4q^{39} + 6q^{41} + 4q^{43} - 2q^{45} + 12q^{47} + 8q^{48} - 12q^{49} - 6q^{51} + 16q^{52} + 6q^{53} + 4q^{57} - 36q^{58} + 6q^{59} - 8q^{60} + 16q^{61} + 24q^{62} - 2q^{63} - 16q^{64} - 4q^{65} - 12q^{66} - 14q^{67} - 24q^{68} - 2q^{69} + 6q^{71} + 4q^{73} - 24q^{74} + 2q^{75} + 16q^{76} - 12q^{78} - 8q^{79} - 8q^{80} + 2q^{81} - 12q^{82} + 6q^{83} - 8q^{84} + 6q^{85} + 6q^{87} - 24q^{88} - 24q^{89} - 4q^{91} - 8q^{92} + 10q^{93} - 12q^{94} - 4q^{95} + 16q^{97} + O(q^{100})$$

## 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
 −2.44949 2.44949
−2.44949 1.00000 4.00000 −1.00000 −2.44949 −1.00000 −4.89898 1.00000 2.44949
1.2 2.44949 1.00000 4.00000 −1.00000 2.44949 −1.00000 4.89898 1.00000 −2.44949
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$5$$ $$1$$
$$23$$ $$1$$

## 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 345.2.a.i 2
3.b odd 2 1 1035.2.a.k 2
4.b odd 2 1 5520.2.a.bi 2
5.b even 2 1 1725.2.a.y 2
5.c odd 4 2 1725.2.b.m 4
15.d odd 2 1 5175.2.a.bl 2
23.b odd 2 1 7935.2.a.t 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
345.2.a.i 2 1.a even 1 1 trivial
1035.2.a.k 2 3.b odd 2 1
1725.2.a.y 2 5.b even 2 1
1725.2.b.m 4 5.c odd 4 2
5175.2.a.bl 2 15.d odd 2 1
5520.2.a.bi 2 4.b odd 2 1
7935.2.a.t 2 23.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(345))$$:

 $$T_{2}^{2} - 6$$ $$T_{7} + 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-6 + T^{2}$$
$3$ $$( -1 + T )^{2}$$
$5$ $$( 1 + T )^{2}$$
$7$ $$( 1 + T )^{2}$$
$11$ $$-6 + T^{2}$$
$13$ $$-2 - 4 T + T^{2}$$
$17$ $$3 + 6 T + T^{2}$$
$19$ $$-2 - 4 T + T^{2}$$
$23$ $$( 1 + T )^{2}$$
$29$ $$-45 - 6 T + T^{2}$$
$31$ $$1 - 10 T + T^{2}$$
$37$ $$-23 + 2 T + T^{2}$$
$41$ $$3 - 6 T + T^{2}$$
$43$ $$( -2 + T )^{2}$$
$47$ $$30 - 12 T + T^{2}$$
$53$ $$3 - 6 T + T^{2}$$
$59$ $$-45 - 6 T + T^{2}$$
$61$ $$10 - 16 T + T^{2}$$
$67$ $$( 7 + T )^{2}$$
$71$ $$-45 - 6 T + T^{2}$$
$73$ $$-50 - 4 T + T^{2}$$
$79$ $$( 4 + T )^{2}$$
$83$ $$-141 - 6 T + T^{2}$$
$89$ $$120 + 24 T + T^{2}$$
$97$ $$40 - 16 T + T^{2}$$