# Properties

 Label 418.2.e.d Level $418$ Weight $2$ Character orbit 418.e Analytic conductor $3.338$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$418 = 2 \cdot 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 418.e (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.33774680449$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\zeta_{6} - 1) q^{2} + ( - 2 \zeta_{6} + 2) q^{3} - \zeta_{6} q^{4} + 2 \zeta_{6} q^{6} + 2 q^{7} + q^{8} - \zeta_{6} q^{9} +O(q^{10})$$ q + (z - 1) * q^2 + (-2*z + 2) * q^3 - z * q^4 + 2*z * q^6 + 2 * q^7 + q^8 - z * q^9 $$q + (\zeta_{6} - 1) q^{2} + ( - 2 \zeta_{6} + 2) q^{3} - \zeta_{6} q^{4} + 2 \zeta_{6} q^{6} + 2 q^{7} + q^{8} - \zeta_{6} q^{9} + q^{11} - 2 q^{12} + \zeta_{6} q^{13} + (2 \zeta_{6} - 2) q^{14} + (\zeta_{6} - 1) q^{16} + ( - 6 \zeta_{6} + 6) q^{17} + q^{18} + ( - 3 \zeta_{6} - 2) q^{19} + ( - 4 \zeta_{6} + 4) q^{21} + (\zeta_{6} - 1) q^{22} + ( - 2 \zeta_{6} + 2) q^{24} + 5 \zeta_{6} q^{25} - q^{26} + 4 q^{27} - 2 \zeta_{6} q^{28} - 9 \zeta_{6} q^{29} + 8 q^{31} - \zeta_{6} q^{32} + ( - 2 \zeta_{6} + 2) q^{33} + 6 \zeta_{6} q^{34} + (\zeta_{6} - 1) q^{36} - 10 q^{37} + ( - 2 \zeta_{6} + 5) q^{38} + 2 q^{39} + ( - 6 \zeta_{6} + 6) q^{41} + 4 \zeta_{6} q^{42} + (5 \zeta_{6} - 5) q^{43} - \zeta_{6} q^{44} + 3 \zeta_{6} q^{47} + 2 \zeta_{6} q^{48} - 3 q^{49} - 5 q^{50} - 12 \zeta_{6} q^{51} + ( - \zeta_{6} + 1) q^{52} + 6 \zeta_{6} q^{53} + (4 \zeta_{6} - 4) q^{54} + 2 q^{56} + (4 \zeta_{6} - 10) q^{57} + 9 q^{58} + (6 \zeta_{6} - 6) q^{59} + 13 \zeta_{6} q^{61} + (8 \zeta_{6} - 8) q^{62} - 2 \zeta_{6} q^{63} + q^{64} + 2 \zeta_{6} q^{66} + 10 \zeta_{6} q^{67} - 6 q^{68} + (9 \zeta_{6} - 9) q^{71} - \zeta_{6} q^{72} + ( - 4 \zeta_{6} + 4) q^{73} + ( - 10 \zeta_{6} + 10) q^{74} + 10 q^{75} + (5 \zeta_{6} - 3) q^{76} + 2 q^{77} + (2 \zeta_{6} - 2) q^{78} + (8 \zeta_{6} - 8) q^{79} + ( - 11 \zeta_{6} + 11) q^{81} + 6 \zeta_{6} q^{82} - 9 q^{83} - 4 q^{84} - 5 \zeta_{6} q^{86} - 18 q^{87} + q^{88} - 15 \zeta_{6} q^{89} + 2 \zeta_{6} q^{91} + ( - 16 \zeta_{6} + 16) q^{93} - 3 q^{94} - 2 q^{96} + ( - \zeta_{6} + 1) q^{97} + ( - 3 \zeta_{6} + 3) q^{98} - \zeta_{6} q^{99} +O(q^{100})$$ q + (z - 1) * q^2 + (-2*z + 2) * q^3 - z * q^4 + 2*z * q^6 + 2 * q^7 + q^8 - z * q^9 + q^11 - 2 * q^12 + z * q^13 + (2*z - 2) * q^14 + (z - 1) * q^16 + (-6*z + 6) * q^17 + q^18 + (-3*z - 2) * q^19 + (-4*z + 4) * q^21 + (z - 1) * q^22 + (-2*z + 2) * q^24 + 5*z * q^25 - q^26 + 4 * q^27 - 2*z * q^28 - 9*z * q^29 + 8 * q^31 - z * q^32 + (-2*z + 2) * q^33 + 6*z * q^34 + (z - 1) * q^36 - 10 * q^37 + (-2*z + 5) * q^38 + 2 * q^39 + (-6*z + 6) * q^41 + 4*z * q^42 + (5*z - 5) * q^43 - z * q^44 + 3*z * q^47 + 2*z * q^48 - 3 * q^49 - 5 * q^50 - 12*z * q^51 + (-z + 1) * q^52 + 6*z * q^53 + (4*z - 4) * q^54 + 2 * q^56 + (4*z - 10) * q^57 + 9 * q^58 + (6*z - 6) * q^59 + 13*z * q^61 + (8*z - 8) * q^62 - 2*z * q^63 + q^64 + 2*z * q^66 + 10*z * q^67 - 6 * q^68 + (9*z - 9) * q^71 - z * q^72 + (-4*z + 4) * q^73 + (-10*z + 10) * q^74 + 10 * q^75 + (5*z - 3) * q^76 + 2 * q^77 + (2*z - 2) * q^78 + (8*z - 8) * q^79 + (-11*z + 11) * q^81 + 6*z * q^82 - 9 * q^83 - 4 * q^84 - 5*z * q^86 - 18 * q^87 + q^88 - 15*z * q^89 + 2*z * q^91 + (-16*z + 16) * q^93 - 3 * q^94 - 2 * q^96 + (-z + 1) * q^97 + (-3*z + 3) * q^98 - z * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} + 2 q^{3} - q^{4} + 2 q^{6} + 4 q^{7} + 2 q^{8} - q^{9}+O(q^{10})$$ 2 * q - q^2 + 2 * q^3 - q^4 + 2 * q^6 + 4 * q^7 + 2 * q^8 - q^9 $$2 q - q^{2} + 2 q^{3} - q^{4} + 2 q^{6} + 4 q^{7} + 2 q^{8} - q^{9} + 2 q^{11} - 4 q^{12} + q^{13} - 2 q^{14} - q^{16} + 6 q^{17} + 2 q^{18} - 7 q^{19} + 4 q^{21} - q^{22} + 2 q^{24} + 5 q^{25} - 2 q^{26} + 8 q^{27} - 2 q^{28} - 9 q^{29} + 16 q^{31} - q^{32} + 2 q^{33} + 6 q^{34} - q^{36} - 20 q^{37} + 8 q^{38} + 4 q^{39} + 6 q^{41} + 4 q^{42} - 5 q^{43} - q^{44} + 3 q^{47} + 2 q^{48} - 6 q^{49} - 10 q^{50} - 12 q^{51} + q^{52} + 6 q^{53} - 4 q^{54} + 4 q^{56} - 16 q^{57} + 18 q^{58} - 6 q^{59} + 13 q^{61} - 8 q^{62} - 2 q^{63} + 2 q^{64} + 2 q^{66} + 10 q^{67} - 12 q^{68} - 9 q^{71} - q^{72} + 4 q^{73} + 10 q^{74} + 20 q^{75} - q^{76} + 4 q^{77} - 2 q^{78} - 8 q^{79} + 11 q^{81} + 6 q^{82} - 18 q^{83} - 8 q^{84} - 5 q^{86} - 36 q^{87} + 2 q^{88} - 15 q^{89} + 2 q^{91} + 16 q^{93} - 6 q^{94} - 4 q^{96} + q^{97} + 3 q^{98} - q^{99}+O(q^{100})$$ 2 * q - q^2 + 2 * q^3 - q^4 + 2 * q^6 + 4 * q^7 + 2 * q^8 - q^9 + 2 * q^11 - 4 * q^12 + q^13 - 2 * q^14 - q^16 + 6 * q^17 + 2 * q^18 - 7 * q^19 + 4 * q^21 - q^22 + 2 * q^24 + 5 * q^25 - 2 * q^26 + 8 * q^27 - 2 * q^28 - 9 * q^29 + 16 * q^31 - q^32 + 2 * q^33 + 6 * q^34 - q^36 - 20 * q^37 + 8 * q^38 + 4 * q^39 + 6 * q^41 + 4 * q^42 - 5 * q^43 - q^44 + 3 * q^47 + 2 * q^48 - 6 * q^49 - 10 * q^50 - 12 * q^51 + q^52 + 6 * q^53 - 4 * q^54 + 4 * q^56 - 16 * q^57 + 18 * q^58 - 6 * q^59 + 13 * q^61 - 8 * q^62 - 2 * q^63 + 2 * q^64 + 2 * q^66 + 10 * q^67 - 12 * q^68 - 9 * q^71 - q^72 + 4 * q^73 + 10 * q^74 + 20 * q^75 - q^76 + 4 * q^77 - 2 * q^78 - 8 * q^79 + 11 * q^81 + 6 * q^82 - 18 * q^83 - 8 * q^84 - 5 * q^86 - 36 * q^87 + 2 * q^88 - 15 * q^89 + 2 * q^91 + 16 * q^93 - 6 * q^94 - 4 * q^96 + q^97 + 3 * q^98 - q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/418\mathbb{Z}\right)^\times$$.

 $$n$$ $$287$$ $$343$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
45.1
 0.5 − 0.866025i 0.5 + 0.866025i
−0.500000 0.866025i 1.00000 + 1.73205i −0.500000 + 0.866025i 0 1.00000 1.73205i 2.00000 1.00000 −0.500000 + 0.866025i 0
353.1 −0.500000 + 0.866025i 1.00000 1.73205i −0.500000 0.866025i 0 1.00000 + 1.73205i 2.00000 1.00000 −0.500000 0.866025i 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 418.2.e.d 2
19.c even 3 1 inner 418.2.e.d 2
19.c even 3 1 7942.2.a.l 1
19.d odd 6 1 7942.2.a.j 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.e.d 2 1.a even 1 1 trivial
418.2.e.d 2 19.c even 3 1 inner
7942.2.a.j 1 19.d odd 6 1
7942.2.a.l 1 19.c even 3 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(418, [\chi])$$:

 $$T_{3}^{2} - 2T_{3} + 4$$ T3^2 - 2*T3 + 4 $$T_{5}$$ T5

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T + 1$$
$3$ $$T^{2} - 2T + 4$$
$5$ $$T^{2}$$
$7$ $$(T - 2)^{2}$$
$11$ $$(T - 1)^{2}$$
$13$ $$T^{2} - T + 1$$
$17$ $$T^{2} - 6T + 36$$
$19$ $$T^{2} + 7T + 19$$
$23$ $$T^{2}$$
$29$ $$T^{2} + 9T + 81$$
$31$ $$(T - 8)^{2}$$
$37$ $$(T + 10)^{2}$$
$41$ $$T^{2} - 6T + 36$$
$43$ $$T^{2} + 5T + 25$$
$47$ $$T^{2} - 3T + 9$$
$53$ $$T^{2} - 6T + 36$$
$59$ $$T^{2} + 6T + 36$$
$61$ $$T^{2} - 13T + 169$$
$67$ $$T^{2} - 10T + 100$$
$71$ $$T^{2} + 9T + 81$$
$73$ $$T^{2} - 4T + 16$$
$79$ $$T^{2} + 8T + 64$$
$83$ $$(T + 9)^{2}$$
$89$ $$T^{2} + 15T + 225$$
$97$ $$T^{2} - T + 1$$