# Properties

 Label 2310.2.a.bb Level $2310$ Weight $2$ Character orbit 2310.a Self dual yes Analytic conductor $18.445$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$2310 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2310.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$18.4454428669$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} + q^{7} + q^{8} + q^{9} +O(q^{10})$$ $$q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} + q^{7} + q^{8} + q^{9} - q^{10} + q^{11} - q^{12} + \beta q^{13} + q^{14} + q^{15} + q^{16} -\beta q^{17} + q^{18} - q^{20} - q^{21} + q^{22} + ( 2 - \beta ) q^{23} - q^{24} + q^{25} + \beta q^{26} - q^{27} + q^{28} + 2 q^{29} + q^{30} + 2 \beta q^{31} + q^{32} - q^{33} -\beta q^{34} - q^{35} + q^{36} + ( 4 + \beta ) q^{37} -\beta q^{39} - q^{40} + ( -2 - 2 \beta ) q^{41} - q^{42} + ( 4 - 2 \beta ) q^{43} + q^{44} - q^{45} + ( 2 - \beta ) q^{46} + 2 \beta q^{47} - q^{48} + q^{49} + q^{50} + \beta q^{51} + \beta q^{52} + ( 6 + 2 \beta ) q^{53} - q^{54} - q^{55} + q^{56} + 2 q^{58} + ( 4 - 2 \beta ) q^{59} + q^{60} + ( 2 - 2 \beta ) q^{61} + 2 \beta q^{62} + q^{63} + q^{64} -\beta q^{65} - q^{66} + ( 6 + \beta ) q^{67} -\beta q^{68} + ( -2 + \beta ) q^{69} - q^{70} -4 q^{71} + q^{72} + ( 6 - 2 \beta ) q^{73} + ( 4 + \beta ) q^{74} - q^{75} + q^{77} -\beta q^{78} + 8 q^{79} - q^{80} + q^{81} + ( -2 - 2 \beta ) q^{82} + 2 \beta q^{83} - q^{84} + \beta q^{85} + ( 4 - 2 \beta ) q^{86} -2 q^{87} + q^{88} + \beta q^{89} - q^{90} + \beta q^{91} + ( 2 - \beta ) q^{92} -2 \beta q^{93} + 2 \beta q^{94} - q^{96} + ( 2 - 4 \beta ) q^{97} + q^{98} + q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} + 2 q^{7} + 2 q^{8} + 2 q^{9} + O(q^{10})$$ $$2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} + 2 q^{7} + 2 q^{8} + 2 q^{9} - 2 q^{10} + 2 q^{11} - 2 q^{12} + 2 q^{14} + 2 q^{15} + 2 q^{16} + 2 q^{18} - 2 q^{20} - 2 q^{21} + 2 q^{22} + 4 q^{23} - 2 q^{24} + 2 q^{25} - 2 q^{27} + 2 q^{28} + 4 q^{29} + 2 q^{30} + 2 q^{32} - 2 q^{33} - 2 q^{35} + 2 q^{36} + 8 q^{37} - 2 q^{40} - 4 q^{41} - 2 q^{42} + 8 q^{43} + 2 q^{44} - 2 q^{45} + 4 q^{46} - 2 q^{48} + 2 q^{49} + 2 q^{50} + 12 q^{53} - 2 q^{54} - 2 q^{55} + 2 q^{56} + 4 q^{58} + 8 q^{59} + 2 q^{60} + 4 q^{61} + 2 q^{63} + 2 q^{64} - 2 q^{66} + 12 q^{67} - 4 q^{69} - 2 q^{70} - 8 q^{71} + 2 q^{72} + 12 q^{73} + 8 q^{74} - 2 q^{75} + 2 q^{77} + 16 q^{79} - 2 q^{80} + 2 q^{81} - 4 q^{82} - 2 q^{84} + 8 q^{86} - 4 q^{87} + 2 q^{88} - 2 q^{90} + 4 q^{92} - 2 q^{96} + 4 q^{97} + 2 q^{98} + 2 q^{99} + O(q^{100})$$

## 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
 −1.73205 1.73205
1.00000 −1.00000 1.00000 −1.00000 −1.00000 1.00000 1.00000 1.00000 −1.00000
1.2 1.00000 −1.00000 1.00000 −1.00000 −1.00000 1.00000 1.00000 1.00000 −1.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$1$$
$$5$$ $$1$$
$$7$$ $$-1$$
$$11$$ $$-1$$

## 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 2310.2.a.bb 2
3.b odd 2 1 6930.2.a.bt 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2310.2.a.bb 2 1.a even 1 1 trivial
6930.2.a.bt 2 3.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2310))$$:

 $$T_{13}^{2} - 12$$ $$T_{17}^{2} - 12$$ $$T_{19}$$ $$T_{23}^{2} - 4 T_{23} - 8$$ $$T_{29} - 2$$ $$T_{31}^{2} - 48$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{2}$$
$3$ $$( 1 + T )^{2}$$
$5$ $$( 1 + T )^{2}$$
$7$ $$( -1 + T )^{2}$$
$11$ $$( -1 + T )^{2}$$
$13$ $$-12 + T^{2}$$
$17$ $$-12 + T^{2}$$
$19$ $$T^{2}$$
$23$ $$-8 - 4 T + T^{2}$$
$29$ $$( -2 + T )^{2}$$
$31$ $$-48 + T^{2}$$
$37$ $$4 - 8 T + T^{2}$$
$41$ $$-44 + 4 T + T^{2}$$
$43$ $$-32 - 8 T + T^{2}$$
$47$ $$-48 + T^{2}$$
$53$ $$-12 - 12 T + T^{2}$$
$59$ $$-32 - 8 T + T^{2}$$
$61$ $$-44 - 4 T + T^{2}$$
$67$ $$24 - 12 T + T^{2}$$
$71$ $$( 4 + T )^{2}$$
$73$ $$-12 - 12 T + T^{2}$$
$79$ $$( -8 + T )^{2}$$
$83$ $$-48 + T^{2}$$
$89$ $$-12 + T^{2}$$
$97$ $$-188 - 4 T + T^{2}$$
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