# Properties

 Label 418.2.a.f Level $418$ Weight $2$ Character orbit 418.a Self dual yes Analytic conductor $3.338$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$418 = 2 \cdot 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 418.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$3.33774680449$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{21})$$ Defining polynomial: $$x^{2} - x - 5$$ x^2 - x - 5 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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 $$\beta = \frac{1}{2}(1 + \sqrt{21})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} - \beta q^{3} + q^{4} + ( - \beta + 2) q^{5} - \beta q^{6} + (\beta - 3) q^{7} + q^{8} + (\beta + 2) q^{9} +O(q^{10})$$ q + q^2 - b * q^3 + q^4 + (-b + 2) * q^5 - b * q^6 + (b - 3) * q^7 + q^8 + (b + 2) * q^9 $$q + q^{2} - \beta q^{3} + q^{4} + ( - \beta + 2) q^{5} - \beta q^{6} + (\beta - 3) q^{7} + q^{8} + (\beta + 2) q^{9} + ( - \beta + 2) q^{10} + q^{11} - \beta q^{12} + (\beta + 3) q^{13} + (\beta - 3) q^{14} + ( - \beta + 5) q^{15} + q^{16} + (2 \beta - 4) q^{17} + (\beta + 2) q^{18} + q^{19} + ( - \beta + 2) q^{20} + (2 \beta - 5) q^{21} + q^{22} + (2 \beta + 2) q^{23} - \beta q^{24} + ( - 3 \beta + 4) q^{25} + (\beta + 3) q^{26} - 5 q^{27} + (\beta - 3) q^{28} + (\beta + 4) q^{29} + ( - \beta + 5) q^{30} + (\beta - 9) q^{31} + q^{32} - \beta q^{33} + (2 \beta - 4) q^{34} + (4 \beta - 11) q^{35} + (\beta + 2) q^{36} + 8 q^{37} + q^{38} + ( - 4 \beta - 5) q^{39} + ( - \beta + 2) q^{40} + (\beta + 1) q^{41} + (2 \beta - 5) q^{42} + ( - 3 \beta + 2) q^{43} + q^{44} + ( - \beta - 1) q^{45} + (2 \beta + 2) q^{46} + ( - 4 \beta + 2) q^{47} - \beta q^{48} + ( - 5 \beta + 7) q^{49} + ( - 3 \beta + 4) q^{50} + (2 \beta - 10) q^{51} + (\beta + 3) q^{52} + (2 \beta + 2) q^{53} - 5 q^{54} + ( - \beta + 2) q^{55} + (\beta - 3) q^{56} - \beta q^{57} + (\beta + 4) q^{58} + ( - \beta + 5) q^{60} + 2 q^{61} + (\beta - 9) q^{62} - q^{63} + q^{64} + ( - 2 \beta + 1) q^{65} - \beta q^{66} + (\beta - 3) q^{67} + (2 \beta - 4) q^{68} + ( - 4 \beta - 10) q^{69} + (4 \beta - 11) q^{70} + (3 \beta - 6) q^{71} + (\beta + 2) q^{72} + ( - 2 \beta + 6) q^{73} + 8 q^{74} + ( - \beta + 15) q^{75} + q^{76} + (\beta - 3) q^{77} + ( - 4 \beta - 5) q^{78} + (2 \beta - 2) q^{79} + ( - \beta + 2) q^{80} + (2 \beta - 6) q^{81} + (\beta + 1) q^{82} + (\beta - 2) q^{83} + (2 \beta - 5) q^{84} + (6 \beta - 18) q^{85} + ( - 3 \beta + 2) q^{86} + ( - 5 \beta - 5) q^{87} + q^{88} + ( - 2 \beta - 8) q^{89} + ( - \beta - 1) q^{90} + (\beta - 4) q^{91} + (2 \beta + 2) q^{92} + (8 \beta - 5) q^{93} + ( - 4 \beta + 2) q^{94} + ( - \beta + 2) q^{95} - \beta q^{96} + 8 q^{97} + ( - 5 \beta + 7) q^{98} + (\beta + 2) q^{99} +O(q^{100})$$ q + q^2 - b * q^3 + q^4 + (-b + 2) * q^5 - b * q^6 + (b - 3) * q^7 + q^8 + (b + 2) * q^9 + (-b + 2) * q^10 + q^11 - b * q^12 + (b + 3) * q^13 + (b - 3) * q^14 + (-b + 5) * q^15 + q^16 + (2*b - 4) * q^17 + (b + 2) * q^18 + q^19 + (-b + 2) * q^20 + (2*b - 5) * q^21 + q^22 + (2*b + 2) * q^23 - b * q^24 + (-3*b + 4) * q^25 + (b + 3) * q^26 - 5 * q^27 + (b - 3) * q^28 + (b + 4) * q^29 + (-b + 5) * q^30 + (b - 9) * q^31 + q^32 - b * q^33 + (2*b - 4) * q^34 + (4*b - 11) * q^35 + (b + 2) * q^36 + 8 * q^37 + q^38 + (-4*b - 5) * q^39 + (-b + 2) * q^40 + (b + 1) * q^41 + (2*b - 5) * q^42 + (-3*b + 2) * q^43 + q^44 + (-b - 1) * q^45 + (2*b + 2) * q^46 + (-4*b + 2) * q^47 - b * q^48 + (-5*b + 7) * q^49 + (-3*b + 4) * q^50 + (2*b - 10) * q^51 + (b + 3) * q^52 + (2*b + 2) * q^53 - 5 * q^54 + (-b + 2) * q^55 + (b - 3) * q^56 - b * q^57 + (b + 4) * q^58 + (-b + 5) * q^60 + 2 * q^61 + (b - 9) * q^62 - q^63 + q^64 + (-2*b + 1) * q^65 - b * q^66 + (b - 3) * q^67 + (2*b - 4) * q^68 + (-4*b - 10) * q^69 + (4*b - 11) * q^70 + (3*b - 6) * q^71 + (b + 2) * q^72 + (-2*b + 6) * q^73 + 8 * q^74 + (-b + 15) * q^75 + q^76 + (b - 3) * q^77 + (-4*b - 5) * q^78 + (2*b - 2) * q^79 + (-b + 2) * q^80 + (2*b - 6) * q^81 + (b + 1) * q^82 + (b - 2) * q^83 + (2*b - 5) * q^84 + (6*b - 18) * q^85 + (-3*b + 2) * q^86 + (-5*b - 5) * q^87 + q^88 + (-2*b - 8) * q^89 + (-b - 1) * q^90 + (b - 4) * q^91 + (2*b + 2) * q^92 + (8*b - 5) * q^93 + (-4*b + 2) * q^94 + (-b + 2) * q^95 - b * q^96 + 8 * q^97 + (-5*b + 7) * q^98 + (b + 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} - q^{3} + 2 q^{4} + 3 q^{5} - q^{6} - 5 q^{7} + 2 q^{8} + 5 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 - q^3 + 2 * q^4 + 3 * q^5 - q^6 - 5 * q^7 + 2 * q^8 + 5 * q^9 $$2 q + 2 q^{2} - q^{3} + 2 q^{4} + 3 q^{5} - q^{6} - 5 q^{7} + 2 q^{8} + 5 q^{9} + 3 q^{10} + 2 q^{11} - q^{12} + 7 q^{13} - 5 q^{14} + 9 q^{15} + 2 q^{16} - 6 q^{17} + 5 q^{18} + 2 q^{19} + 3 q^{20} - 8 q^{21} + 2 q^{22} + 6 q^{23} - q^{24} + 5 q^{25} + 7 q^{26} - 10 q^{27} - 5 q^{28} + 9 q^{29} + 9 q^{30} - 17 q^{31} + 2 q^{32} - q^{33} - 6 q^{34} - 18 q^{35} + 5 q^{36} + 16 q^{37} + 2 q^{38} - 14 q^{39} + 3 q^{40} + 3 q^{41} - 8 q^{42} + q^{43} + 2 q^{44} - 3 q^{45} + 6 q^{46} - q^{48} + 9 q^{49} + 5 q^{50} - 18 q^{51} + 7 q^{52} + 6 q^{53} - 10 q^{54} + 3 q^{55} - 5 q^{56} - q^{57} + 9 q^{58} + 9 q^{60} + 4 q^{61} - 17 q^{62} - 2 q^{63} + 2 q^{64} - q^{66} - 5 q^{67} - 6 q^{68} - 24 q^{69} - 18 q^{70} - 9 q^{71} + 5 q^{72} + 10 q^{73} + 16 q^{74} + 29 q^{75} + 2 q^{76} - 5 q^{77} - 14 q^{78} - 2 q^{79} + 3 q^{80} - 10 q^{81} + 3 q^{82} - 3 q^{83} - 8 q^{84} - 30 q^{85} + q^{86} - 15 q^{87} + 2 q^{88} - 18 q^{89} - 3 q^{90} - 7 q^{91} + 6 q^{92} - 2 q^{93} + 3 q^{95} - q^{96} + 16 q^{97} + 9 q^{98} + 5 q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 - q^3 + 2 * q^4 + 3 * q^5 - q^6 - 5 * q^7 + 2 * q^8 + 5 * q^9 + 3 * q^10 + 2 * q^11 - q^12 + 7 * q^13 - 5 * q^14 + 9 * q^15 + 2 * q^16 - 6 * q^17 + 5 * q^18 + 2 * q^19 + 3 * q^20 - 8 * q^21 + 2 * q^22 + 6 * q^23 - q^24 + 5 * q^25 + 7 * q^26 - 10 * q^27 - 5 * q^28 + 9 * q^29 + 9 * q^30 - 17 * q^31 + 2 * q^32 - q^33 - 6 * q^34 - 18 * q^35 + 5 * q^36 + 16 * q^37 + 2 * q^38 - 14 * q^39 + 3 * q^40 + 3 * q^41 - 8 * q^42 + q^43 + 2 * q^44 - 3 * q^45 + 6 * q^46 - q^48 + 9 * q^49 + 5 * q^50 - 18 * q^51 + 7 * q^52 + 6 * q^53 - 10 * q^54 + 3 * q^55 - 5 * q^56 - q^57 + 9 * q^58 + 9 * q^60 + 4 * q^61 - 17 * q^62 - 2 * q^63 + 2 * q^64 - q^66 - 5 * q^67 - 6 * q^68 - 24 * q^69 - 18 * q^70 - 9 * q^71 + 5 * q^72 + 10 * q^73 + 16 * q^74 + 29 * q^75 + 2 * q^76 - 5 * q^77 - 14 * q^78 - 2 * q^79 + 3 * q^80 - 10 * q^81 + 3 * q^82 - 3 * q^83 - 8 * q^84 - 30 * q^85 + q^86 - 15 * q^87 + 2 * q^88 - 18 * q^89 - 3 * q^90 - 7 * q^91 + 6 * q^92 - 2 * q^93 + 3 * q^95 - q^96 + 16 * q^97 + 9 * q^98 + 5 * q^99

## 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
 2.79129 −1.79129
1.00000 −2.79129 1.00000 −0.791288 −2.79129 −0.208712 1.00000 4.79129 −0.791288
1.2 1.00000 1.79129 1.00000 3.79129 1.79129 −4.79129 1.00000 0.208712 3.79129
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$11$$ $$-1$$
$$19$$ $$-1$$

## 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 418.2.a.f 2
3.b odd 2 1 3762.2.a.s 2
4.b odd 2 1 3344.2.a.l 2
11.b odd 2 1 4598.2.a.y 2
19.b odd 2 1 7942.2.a.w 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.a.f 2 1.a even 1 1 trivial
3344.2.a.l 2 4.b odd 2 1
3762.2.a.s 2 3.b odd 2 1
4598.2.a.y 2 11.b odd 2 1
7942.2.a.w 2 19.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + T_{3} - 5$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(418))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{2}$$
$3$ $$T^{2} + T - 5$$
$5$ $$T^{2} - 3T - 3$$
$7$ $$T^{2} + 5T + 1$$
$11$ $$(T - 1)^{2}$$
$13$ $$T^{2} - 7T + 7$$
$17$ $$T^{2} + 6T - 12$$
$19$ $$(T - 1)^{2}$$
$23$ $$T^{2} - 6T - 12$$
$29$ $$T^{2} - 9T + 15$$
$31$ $$T^{2} + 17T + 67$$
$37$ $$(T - 8)^{2}$$
$41$ $$T^{2} - 3T - 3$$
$43$ $$T^{2} - T - 47$$
$47$ $$T^{2} - 84$$
$53$ $$T^{2} - 6T - 12$$
$59$ $$T^{2}$$
$61$ $$(T - 2)^{2}$$
$67$ $$T^{2} + 5T + 1$$
$71$ $$T^{2} + 9T - 27$$
$73$ $$T^{2} - 10T + 4$$
$79$ $$T^{2} + 2T - 20$$
$83$ $$T^{2} + 3T - 3$$
$89$ $$T^{2} + 18T + 60$$
$97$ $$(T - 8)^{2}$$
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