# Properties

 Label 418.2.e.f 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.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{6} + 1) q^{2} + ( - 2 \zeta_{6} + 2) q^{3} - \zeta_{6} q^{4} + ( - 3 \zeta_{6} + 3) q^{5} - 2 \zeta_{6} q^{6} - q^{7} - q^{8} - \zeta_{6} q^{9} +O(q^{10})$$ q + (-z + 1) * q^2 + (-2*z + 2) * q^3 - z * q^4 + (-3*z + 3) * q^5 - 2*z * q^6 - q^7 - q^8 - z * q^9 $$q + ( - \zeta_{6} + 1) q^{2} + ( - 2 \zeta_{6} + 2) q^{3} - \zeta_{6} q^{4} + ( - 3 \zeta_{6} + 3) q^{5} - 2 \zeta_{6} q^{6} - q^{7} - q^{8} - \zeta_{6} q^{9} - 3 \zeta_{6} q^{10} + q^{11} - 2 q^{12} + 4 \zeta_{6} q^{13} + (\zeta_{6} - 1) q^{14} - 6 \zeta_{6} q^{15} + (\zeta_{6} - 1) q^{16} - q^{18} + (3 \zeta_{6} - 5) q^{19} - 3 q^{20} + (2 \zeta_{6} - 2) q^{21} + ( - \zeta_{6} + 1) q^{22} + (2 \zeta_{6} - 2) q^{24} - 4 \zeta_{6} q^{25} + 4 q^{26} + 4 q^{27} + \zeta_{6} q^{28} + 6 \zeta_{6} q^{29} - 6 q^{30} - 4 q^{31} + \zeta_{6} q^{32} + ( - 2 \zeta_{6} + 2) q^{33} + (3 \zeta_{6} - 3) q^{35} + (\zeta_{6} - 1) q^{36} - q^{37} + (5 \zeta_{6} - 2) q^{38} + 8 q^{39} + (3 \zeta_{6} - 3) q^{40} + ( - 6 \zeta_{6} + 6) q^{41} + 2 \zeta_{6} q^{42} + ( - 4 \zeta_{6} + 4) q^{43} - \zeta_{6} q^{44} - 3 q^{45} - 6 \zeta_{6} q^{47} + 2 \zeta_{6} q^{48} - 6 q^{49} - 4 q^{50} + ( - 4 \zeta_{6} + 4) q^{52} - 3 \zeta_{6} q^{53} + ( - 4 \zeta_{6} + 4) q^{54} + ( - 3 \zeta_{6} + 3) q^{55} + q^{56} + (10 \zeta_{6} - 4) q^{57} + 6 q^{58} + (6 \zeta_{6} - 6) q^{60} + 4 \zeta_{6} q^{61} + (4 \zeta_{6} - 4) q^{62} + \zeta_{6} q^{63} + q^{64} + 12 q^{65} - 2 \zeta_{6} q^{66} + 4 \zeta_{6} q^{67} + 3 \zeta_{6} q^{70} + ( - 6 \zeta_{6} + 6) q^{71} + \zeta_{6} q^{72} + (2 \zeta_{6} - 2) q^{73} + (\zeta_{6} - 1) q^{74} - 8 q^{75} + (2 \zeta_{6} + 3) q^{76} - q^{77} + ( - 8 \zeta_{6} + 8) q^{78} + (11 \zeta_{6} - 11) q^{79} + 3 \zeta_{6} q^{80} + ( - 11 \zeta_{6} + 11) q^{81} - 6 \zeta_{6} q^{82} - 3 q^{83} + 2 q^{84} - 4 \zeta_{6} q^{86} + 12 q^{87} - q^{88} + 18 \zeta_{6} q^{89} + (3 \zeta_{6} - 3) q^{90} - 4 \zeta_{6} q^{91} + (8 \zeta_{6} - 8) q^{93} - 6 q^{94} + (15 \zeta_{6} - 6) q^{95} + 2 q^{96} + ( - 19 \zeta_{6} + 19) q^{97} + (6 \zeta_{6} - 6) q^{98} - \zeta_{6} q^{99} +O(q^{100})$$ q + (-z + 1) * q^2 + (-2*z + 2) * q^3 - z * q^4 + (-3*z + 3) * q^5 - 2*z * q^6 - q^7 - q^8 - z * q^9 - 3*z * q^10 + q^11 - 2 * q^12 + 4*z * q^13 + (z - 1) * q^14 - 6*z * q^15 + (z - 1) * q^16 - q^18 + (3*z - 5) * q^19 - 3 * q^20 + (2*z - 2) * q^21 + (-z + 1) * q^22 + (2*z - 2) * q^24 - 4*z * q^25 + 4 * q^26 + 4 * q^27 + z * q^28 + 6*z * q^29 - 6 * q^30 - 4 * q^31 + z * q^32 + (-2*z + 2) * q^33 + (3*z - 3) * q^35 + (z - 1) * q^36 - q^37 + (5*z - 2) * q^38 + 8 * q^39 + (3*z - 3) * q^40 + (-6*z + 6) * q^41 + 2*z * q^42 + (-4*z + 4) * q^43 - z * q^44 - 3 * q^45 - 6*z * q^47 + 2*z * q^48 - 6 * q^49 - 4 * q^50 + (-4*z + 4) * q^52 - 3*z * q^53 + (-4*z + 4) * q^54 + (-3*z + 3) * q^55 + q^56 + (10*z - 4) * q^57 + 6 * q^58 + (6*z - 6) * q^60 + 4*z * q^61 + (4*z - 4) * q^62 + z * q^63 + q^64 + 12 * q^65 - 2*z * q^66 + 4*z * q^67 + 3*z * q^70 + (-6*z + 6) * q^71 + z * q^72 + (2*z - 2) * q^73 + (z - 1) * q^74 - 8 * q^75 + (2*z + 3) * q^76 - q^77 + (-8*z + 8) * q^78 + (11*z - 11) * q^79 + 3*z * q^80 + (-11*z + 11) * q^81 - 6*z * q^82 - 3 * q^83 + 2 * q^84 - 4*z * q^86 + 12 * q^87 - q^88 + 18*z * q^89 + (3*z - 3) * q^90 - 4*z * q^91 + (8*z - 8) * q^93 - 6 * q^94 + (15*z - 6) * q^95 + 2 * q^96 + (-19*z + 19) * q^97 + (6*z - 6) * q^98 - z * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} + 2 q^{3} - q^{4} + 3 q^{5} - 2 q^{6} - 2 q^{7} - 2 q^{8} - q^{9}+O(q^{10})$$ 2 * q + q^2 + 2 * q^3 - q^4 + 3 * q^5 - 2 * q^6 - 2 * q^7 - 2 * q^8 - q^9 $$2 q + q^{2} + 2 q^{3} - q^{4} + 3 q^{5} - 2 q^{6} - 2 q^{7} - 2 q^{8} - q^{9} - 3 q^{10} + 2 q^{11} - 4 q^{12} + 4 q^{13} - q^{14} - 6 q^{15} - q^{16} - 2 q^{18} - 7 q^{19} - 6 q^{20} - 2 q^{21} + q^{22} - 2 q^{24} - 4 q^{25} + 8 q^{26} + 8 q^{27} + q^{28} + 6 q^{29} - 12 q^{30} - 8 q^{31} + q^{32} + 2 q^{33} - 3 q^{35} - q^{36} - 2 q^{37} + q^{38} + 16 q^{39} - 3 q^{40} + 6 q^{41} + 2 q^{42} + 4 q^{43} - q^{44} - 6 q^{45} - 6 q^{47} + 2 q^{48} - 12 q^{49} - 8 q^{50} + 4 q^{52} - 3 q^{53} + 4 q^{54} + 3 q^{55} + 2 q^{56} + 2 q^{57} + 12 q^{58} - 6 q^{60} + 4 q^{61} - 4 q^{62} + q^{63} + 2 q^{64} + 24 q^{65} - 2 q^{66} + 4 q^{67} + 3 q^{70} + 6 q^{71} + q^{72} - 2 q^{73} - q^{74} - 16 q^{75} + 8 q^{76} - 2 q^{77} + 8 q^{78} - 11 q^{79} + 3 q^{80} + 11 q^{81} - 6 q^{82} - 6 q^{83} + 4 q^{84} - 4 q^{86} + 24 q^{87} - 2 q^{88} + 18 q^{89} - 3 q^{90} - 4 q^{91} - 8 q^{93} - 12 q^{94} + 3 q^{95} + 4 q^{96} + 19 q^{97} - 6 q^{98} - q^{99}+O(q^{100})$$ 2 * q + q^2 + 2 * q^3 - q^4 + 3 * q^5 - 2 * q^6 - 2 * q^7 - 2 * q^8 - q^9 - 3 * q^10 + 2 * q^11 - 4 * q^12 + 4 * q^13 - q^14 - 6 * q^15 - q^16 - 2 * q^18 - 7 * q^19 - 6 * q^20 - 2 * q^21 + q^22 - 2 * q^24 - 4 * q^25 + 8 * q^26 + 8 * q^27 + q^28 + 6 * q^29 - 12 * q^30 - 8 * q^31 + q^32 + 2 * q^33 - 3 * q^35 - q^36 - 2 * q^37 + q^38 + 16 * q^39 - 3 * q^40 + 6 * q^41 + 2 * q^42 + 4 * q^43 - q^44 - 6 * q^45 - 6 * q^47 + 2 * q^48 - 12 * q^49 - 8 * q^50 + 4 * q^52 - 3 * q^53 + 4 * q^54 + 3 * q^55 + 2 * q^56 + 2 * q^57 + 12 * q^58 - 6 * q^60 + 4 * q^61 - 4 * q^62 + q^63 + 2 * q^64 + 24 * q^65 - 2 * q^66 + 4 * q^67 + 3 * q^70 + 6 * q^71 + q^72 - 2 * q^73 - q^74 - 16 * q^75 + 8 * q^76 - 2 * q^77 + 8 * q^78 - 11 * q^79 + 3 * q^80 + 11 * q^81 - 6 * q^82 - 6 * q^83 + 4 * q^84 - 4 * q^86 + 24 * q^87 - 2 * q^88 + 18 * q^89 - 3 * q^90 - 4 * q^91 - 8 * q^93 - 12 * q^94 + 3 * q^95 + 4 * q^96 + 19 * q^97 - 6 * 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 1.50000 + 2.59808i −1.00000 + 1.73205i −1.00000 −1.00000 −0.500000 + 0.866025i −1.50000 + 2.59808i
353.1 0.500000 0.866025i 1.00000 1.73205i −0.500000 0.866025i 1.50000 2.59808i −1.00000 1.73205i −1.00000 −1.00000 −0.500000 0.866025i −1.50000 2.59808i
 $$n$$: 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.f 2
19.c even 3 1 inner 418.2.e.f 2
19.c even 3 1 7942.2.a.b 1
19.d odd 6 1 7942.2.a.r 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.e.f 2 1.a even 1 1 trivial
418.2.e.f 2 19.c even 3 1 inner
7942.2.a.b 1 19.c even 3 1
7942.2.a.r 1 19.d odd 6 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 $$T_{5}^{2} - 3T_{5} + 9$$ T5^2 - 3*T5 + 9

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T + 1$$
$3$ $$T^{2} - 2T + 4$$
$5$ $$T^{2} - 3T + 9$$
$7$ $$(T + 1)^{2}$$
$11$ $$(T - 1)^{2}$$
$13$ $$T^{2} - 4T + 16$$
$17$ $$T^{2}$$
$19$ $$T^{2} + 7T + 19$$
$23$ $$T^{2}$$
$29$ $$T^{2} - 6T + 36$$
$31$ $$(T + 4)^{2}$$
$37$ $$(T + 1)^{2}$$
$41$ $$T^{2} - 6T + 36$$
$43$ $$T^{2} - 4T + 16$$
$47$ $$T^{2} + 6T + 36$$
$53$ $$T^{2} + 3T + 9$$
$59$ $$T^{2}$$
$61$ $$T^{2} - 4T + 16$$
$67$ $$T^{2} - 4T + 16$$
$71$ $$T^{2} - 6T + 36$$
$73$ $$T^{2} + 2T + 4$$
$79$ $$T^{2} + 11T + 121$$
$83$ $$(T + 3)^{2}$$
$89$ $$T^{2} - 18T + 324$$
$97$ $$T^{2} - 19T + 361$$