# Properties

 Label 2394.2.a.s Level $2394$ Weight $2$ Character orbit 2394.a Self dual yes Analytic conductor $19.116$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$2394 = 2 \cdot 3^{2} \cdot 7 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2394.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$19.1161862439$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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 $$\beta = \sqrt{5}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} + (\beta + 1) q^{5} - q^{7} - q^{8}+O(q^{10})$$ q - q^2 + q^4 + (b + 1) * q^5 - q^7 - q^8 $$q - q^{2} + q^{4} + (\beta + 1) q^{5} - q^{7} - q^{8} + ( - \beta - 1) q^{10} + ( - \beta + 3) q^{11} + (\beta - 1) q^{13} + q^{14} + q^{16} + ( - 3 \beta - 1) q^{17} + q^{19} + (\beta + 1) q^{20} + (\beta - 3) q^{22} + (\beta + 3) q^{23} + (2 \beta + 1) q^{25} + ( - \beta + 1) q^{26} - q^{28} + (2 \beta + 4) q^{29} - 2 q^{31} - q^{32} + (3 \beta + 1) q^{34} + ( - \beta - 1) q^{35} + ( - 2 \beta - 6) q^{37} - q^{38} + ( - \beta - 1) q^{40} + 6 q^{41} + 4 \beta q^{43} + ( - \beta + 3) q^{44} + ( - \beta - 3) q^{46} + 4 \beta q^{47} + q^{49} + ( - 2 \beta - 1) q^{50} + (\beta - 1) q^{52} + (2 \beta + 8) q^{53} + (2 \beta - 2) q^{55} + q^{56} + ( - 2 \beta - 4) q^{58} + 8 q^{59} + ( - 2 \beta + 4) q^{61} + 2 q^{62} + q^{64} + 4 q^{65} + ( - 3 \beta + 7) q^{67} + ( - 3 \beta - 1) q^{68} + (\beta + 1) q^{70} + (2 \beta + 6) q^{71} + ( - 2 \beta - 8) q^{73} + (2 \beta + 6) q^{74} + q^{76} + (\beta - 3) q^{77} + ( - \beta + 3) q^{79} + (\beta + 1) q^{80} - 6 q^{82} - 10 q^{83} + ( - 4 \beta - 16) q^{85} - 4 \beta q^{86} + (\beta - 3) q^{88} + 6 q^{89} + ( - \beta + 1) q^{91} + (\beta + 3) q^{92} - 4 \beta q^{94} + (\beta + 1) q^{95} + ( - \beta + 7) q^{97} - q^{98} +O(q^{100})$$ q - q^2 + q^4 + (b + 1) * q^5 - q^7 - q^8 + (-b - 1) * q^10 + (-b + 3) * q^11 + (b - 1) * q^13 + q^14 + q^16 + (-3*b - 1) * q^17 + q^19 + (b + 1) * q^20 + (b - 3) * q^22 + (b + 3) * q^23 + (2*b + 1) * q^25 + (-b + 1) * q^26 - q^28 + (2*b + 4) * q^29 - 2 * q^31 - q^32 + (3*b + 1) * q^34 + (-b - 1) * q^35 + (-2*b - 6) * q^37 - q^38 + (-b - 1) * q^40 + 6 * q^41 + 4*b * q^43 + (-b + 3) * q^44 + (-b - 3) * q^46 + 4*b * q^47 + q^49 + (-2*b - 1) * q^50 + (b - 1) * q^52 + (2*b + 8) * q^53 + (2*b - 2) * q^55 + q^56 + (-2*b - 4) * q^58 + 8 * q^59 + (-2*b + 4) * q^61 + 2 * q^62 + q^64 + 4 * q^65 + (-3*b + 7) * q^67 + (-3*b - 1) * q^68 + (b + 1) * q^70 + (2*b + 6) * q^71 + (-2*b - 8) * q^73 + (2*b + 6) * q^74 + q^76 + (b - 3) * q^77 + (-b + 3) * q^79 + (b + 1) * q^80 - 6 * q^82 - 10 * q^83 + (-4*b - 16) * q^85 - 4*b * q^86 + (b - 3) * q^88 + 6 * q^89 + (-b + 1) * q^91 + (b + 3) * q^92 - 4*b * q^94 + (b + 1) * q^95 + (-b + 7) * q^97 - q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - 2 q^{7} - 2 q^{8}+O(q^{10})$$ 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^5 - 2 * q^7 - 2 * q^8 $$2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - 2 q^{7} - 2 q^{8} - 2 q^{10} + 6 q^{11} - 2 q^{13} + 2 q^{14} + 2 q^{16} - 2 q^{17} + 2 q^{19} + 2 q^{20} - 6 q^{22} + 6 q^{23} + 2 q^{25} + 2 q^{26} - 2 q^{28} + 8 q^{29} - 4 q^{31} - 2 q^{32} + 2 q^{34} - 2 q^{35} - 12 q^{37} - 2 q^{38} - 2 q^{40} + 12 q^{41} + 6 q^{44} - 6 q^{46} + 2 q^{49} - 2 q^{50} - 2 q^{52} + 16 q^{53} - 4 q^{55} + 2 q^{56} - 8 q^{58} + 16 q^{59} + 8 q^{61} + 4 q^{62} + 2 q^{64} + 8 q^{65} + 14 q^{67} - 2 q^{68} + 2 q^{70} + 12 q^{71} - 16 q^{73} + 12 q^{74} + 2 q^{76} - 6 q^{77} + 6 q^{79} + 2 q^{80} - 12 q^{82} - 20 q^{83} - 32 q^{85} - 6 q^{88} + 12 q^{89} + 2 q^{91} + 6 q^{92} + 2 q^{95} + 14 q^{97} - 2 q^{98}+O(q^{100})$$ 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^5 - 2 * q^7 - 2 * q^8 - 2 * q^10 + 6 * q^11 - 2 * q^13 + 2 * q^14 + 2 * q^16 - 2 * q^17 + 2 * q^19 + 2 * q^20 - 6 * q^22 + 6 * q^23 + 2 * q^25 + 2 * q^26 - 2 * q^28 + 8 * q^29 - 4 * q^31 - 2 * q^32 + 2 * q^34 - 2 * q^35 - 12 * q^37 - 2 * q^38 - 2 * q^40 + 12 * q^41 + 6 * q^44 - 6 * q^46 + 2 * q^49 - 2 * q^50 - 2 * q^52 + 16 * q^53 - 4 * q^55 + 2 * q^56 - 8 * q^58 + 16 * q^59 + 8 * q^61 + 4 * q^62 + 2 * q^64 + 8 * q^65 + 14 * q^67 - 2 * q^68 + 2 * q^70 + 12 * q^71 - 16 * q^73 + 12 * q^74 + 2 * q^76 - 6 * q^77 + 6 * q^79 + 2 * q^80 - 12 * q^82 - 20 * q^83 - 32 * q^85 - 6 * q^88 + 12 * q^89 + 2 * q^91 + 6 * q^92 + 2 * q^95 + 14 * q^97 - 2 * q^98

## 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
 −0.618034 1.61803
−1.00000 0 1.00000 −1.23607 0 −1.00000 −1.00000 0 1.23607
1.2 −1.00000 0 1.00000 3.23607 0 −1.00000 −1.00000 0 −3.23607
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$1$$
$$7$$ $$1$$
$$19$$ $$-1$$

## 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 2394.2.a.s 2
3.b odd 2 1 2394.2.a.v yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2394.2.a.s 2 1.a even 1 1 trivial
2394.2.a.v yes 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2394))$$:

 $$T_{5}^{2} - 2T_{5} - 4$$ T5^2 - 2*T5 - 4 $$T_{11}^{2} - 6T_{11} + 4$$ T11^2 - 6*T11 + 4 $$T_{13}^{2} + 2T_{13} - 4$$ T13^2 + 2*T13 - 4 $$T_{17}^{2} + 2T_{17} - 44$$ T17^2 + 2*T17 - 44

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 2T - 4$$
$7$ $$(T + 1)^{2}$$
$11$ $$T^{2} - 6T + 4$$
$13$ $$T^{2} + 2T - 4$$
$17$ $$T^{2} + 2T - 44$$
$19$ $$(T - 1)^{2}$$
$23$ $$T^{2} - 6T + 4$$
$29$ $$T^{2} - 8T - 4$$
$31$ $$(T + 2)^{2}$$
$37$ $$T^{2} + 12T + 16$$
$41$ $$(T - 6)^{2}$$
$43$ $$T^{2} - 80$$
$47$ $$T^{2} - 80$$
$53$ $$T^{2} - 16T + 44$$
$59$ $$(T - 8)^{2}$$
$61$ $$T^{2} - 8T - 4$$
$67$ $$T^{2} - 14T + 4$$
$71$ $$T^{2} - 12T + 16$$
$73$ $$T^{2} + 16T + 44$$
$79$ $$T^{2} - 6T + 4$$
$83$ $$(T + 10)^{2}$$
$89$ $$(T - 6)^{2}$$
$97$ $$T^{2} - 14T + 44$$
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