# Properties

 Label 418.2.e.c 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.

 $$f(q)$$ $$=$$ $$q + (\zeta_{6} - 1) q^{2} - \zeta_{6} q^{4} + ( - 4 \zeta_{6} + 4) q^{5} - 4 q^{7} + q^{8} + 3 \zeta_{6} q^{9} +O(q^{10})$$ q + (z - 1) * q^2 - z * q^4 + (-4*z + 4) * q^5 - 4 * q^7 + q^8 + 3*z * q^9 $$q + (\zeta_{6} - 1) q^{2} - \zeta_{6} q^{4} + ( - 4 \zeta_{6} + 4) q^{5} - 4 q^{7} + q^{8} + 3 \zeta_{6} q^{9} + 4 \zeta_{6} q^{10} + q^{11} - 7 \zeta_{6} q^{13} + ( - 4 \zeta_{6} + 4) q^{14} + (\zeta_{6} - 1) q^{16} - 3 q^{18} + ( - 5 \zeta_{6} + 2) q^{19} - 4 q^{20} + (\zeta_{6} - 1) q^{22} - 4 \zeta_{6} q^{23} - 11 \zeta_{6} q^{25} + 7 q^{26} + 4 \zeta_{6} q^{28} + 3 \zeta_{6} q^{29} - \zeta_{6} q^{32} + (16 \zeta_{6} - 16) q^{35} + ( - 3 \zeta_{6} + 3) q^{36} + 8 q^{37} + (2 \zeta_{6} + 3) q^{38} + ( - 4 \zeta_{6} + 4) q^{40} + ( - 12 \zeta_{6} + 12) q^{41} + (7 \zeta_{6} - 7) q^{43} - \zeta_{6} q^{44} + 12 q^{45} + 4 q^{46} + 5 \zeta_{6} q^{47} + 9 q^{49} + 11 q^{50} + (7 \zeta_{6} - 7) q^{52} - 6 \zeta_{6} q^{53} + ( - 4 \zeta_{6} + 4) q^{55} - 4 q^{56} - 3 q^{58} + (6 \zeta_{6} - 6) q^{59} + 5 \zeta_{6} q^{61} - 12 \zeta_{6} q^{63} + q^{64} - 28 q^{65} + 8 \zeta_{6} q^{67} - 16 \zeta_{6} q^{70} + ( - 9 \zeta_{6} + 9) q^{71} + 3 \zeta_{6} q^{72} + (10 \zeta_{6} - 10) q^{73} + (8 \zeta_{6} - 8) q^{74} + (3 \zeta_{6} - 5) q^{76} - 4 q^{77} + ( - 6 \zeta_{6} + 6) q^{79} + 4 \zeta_{6} q^{80} + (9 \zeta_{6} - 9) q^{81} + 12 \zeta_{6} q^{82} - 3 q^{83} - 7 \zeta_{6} q^{86} + q^{88} + \zeta_{6} q^{89} + (12 \zeta_{6} - 12) q^{90} + 28 \zeta_{6} q^{91} + (4 \zeta_{6} - 4) q^{92} - 5 q^{94} + ( - 8 \zeta_{6} - 12) q^{95} + ( - 5 \zeta_{6} + 5) q^{97} + (9 \zeta_{6} - 9) q^{98} + 3 \zeta_{6} q^{99} +O(q^{100})$$ q + (z - 1) * q^2 - z * q^4 + (-4*z + 4) * q^5 - 4 * q^7 + q^8 + 3*z * q^9 + 4*z * q^10 + q^11 - 7*z * q^13 + (-4*z + 4) * q^14 + (z - 1) * q^16 - 3 * q^18 + (-5*z + 2) * q^19 - 4 * q^20 + (z - 1) * q^22 - 4*z * q^23 - 11*z * q^25 + 7 * q^26 + 4*z * q^28 + 3*z * q^29 - z * q^32 + (16*z - 16) * q^35 + (-3*z + 3) * q^36 + 8 * q^37 + (2*z + 3) * q^38 + (-4*z + 4) * q^40 + (-12*z + 12) * q^41 + (7*z - 7) * q^43 - z * q^44 + 12 * q^45 + 4 * q^46 + 5*z * q^47 + 9 * q^49 + 11 * q^50 + (7*z - 7) * q^52 - 6*z * q^53 + (-4*z + 4) * q^55 - 4 * q^56 - 3 * q^58 + (6*z - 6) * q^59 + 5*z * q^61 - 12*z * q^63 + q^64 - 28 * q^65 + 8*z * q^67 - 16*z * q^70 + (-9*z + 9) * q^71 + 3*z * q^72 + (10*z - 10) * q^73 + (8*z - 8) * q^74 + (3*z - 5) * q^76 - 4 * q^77 + (-6*z + 6) * q^79 + 4*z * q^80 + (9*z - 9) * q^81 + 12*z * q^82 - 3 * q^83 - 7*z * q^86 + q^88 + z * q^89 + (12*z - 12) * q^90 + 28*z * q^91 + (4*z - 4) * q^92 - 5 * q^94 + (-8*z - 12) * q^95 + (-5*z + 5) * q^97 + (9*z - 9) * q^98 + 3*z * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} - q^{4} + 4 q^{5} - 8 q^{7} + 2 q^{8} + 3 q^{9}+O(q^{10})$$ 2 * q - q^2 - q^4 + 4 * q^5 - 8 * q^7 + 2 * q^8 + 3 * q^9 $$2 q - q^{2} - q^{4} + 4 q^{5} - 8 q^{7} + 2 q^{8} + 3 q^{9} + 4 q^{10} + 2 q^{11} - 7 q^{13} + 4 q^{14} - q^{16} - 6 q^{18} - q^{19} - 8 q^{20} - q^{22} - 4 q^{23} - 11 q^{25} + 14 q^{26} + 4 q^{28} + 3 q^{29} - q^{32} - 16 q^{35} + 3 q^{36} + 16 q^{37} + 8 q^{38} + 4 q^{40} + 12 q^{41} - 7 q^{43} - q^{44} + 24 q^{45} + 8 q^{46} + 5 q^{47} + 18 q^{49} + 22 q^{50} - 7 q^{52} - 6 q^{53} + 4 q^{55} - 8 q^{56} - 6 q^{58} - 6 q^{59} + 5 q^{61} - 12 q^{63} + 2 q^{64} - 56 q^{65} + 8 q^{67} - 16 q^{70} + 9 q^{71} + 3 q^{72} - 10 q^{73} - 8 q^{74} - 7 q^{76} - 8 q^{77} + 6 q^{79} + 4 q^{80} - 9 q^{81} + 12 q^{82} - 6 q^{83} - 7 q^{86} + 2 q^{88} + q^{89} - 12 q^{90} + 28 q^{91} - 4 q^{92} - 10 q^{94} - 32 q^{95} + 5 q^{97} - 9 q^{98} + 3 q^{99}+O(q^{100})$$ 2 * q - q^2 - q^4 + 4 * q^5 - 8 * q^7 + 2 * q^8 + 3 * q^9 + 4 * q^10 + 2 * q^11 - 7 * q^13 + 4 * q^14 - q^16 - 6 * q^18 - q^19 - 8 * q^20 - q^22 - 4 * q^23 - 11 * q^25 + 14 * q^26 + 4 * q^28 + 3 * q^29 - q^32 - 16 * q^35 + 3 * q^36 + 16 * q^37 + 8 * q^38 + 4 * q^40 + 12 * q^41 - 7 * q^43 - q^44 + 24 * q^45 + 8 * q^46 + 5 * q^47 + 18 * q^49 + 22 * q^50 - 7 * q^52 - 6 * q^53 + 4 * q^55 - 8 * q^56 - 6 * q^58 - 6 * q^59 + 5 * q^61 - 12 * q^63 + 2 * q^64 - 56 * q^65 + 8 * q^67 - 16 * q^70 + 9 * q^71 + 3 * q^72 - 10 * q^73 - 8 * q^74 - 7 * q^76 - 8 * q^77 + 6 * q^79 + 4 * q^80 - 9 * q^81 + 12 * q^82 - 6 * q^83 - 7 * q^86 + 2 * q^88 + q^89 - 12 * q^90 + 28 * q^91 - 4 * q^92 - 10 * q^94 - 32 * q^95 + 5 * q^97 - 9 * q^98 + 3 * 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 0 −0.500000 + 0.866025i 2.00000 + 3.46410i 0 −4.00000 1.00000 1.50000 2.59808i 2.00000 3.46410i
353.1 −0.500000 + 0.866025i 0 −0.500000 0.866025i 2.00000 3.46410i 0 −4.00000 1.00000 1.50000 + 2.59808i 2.00000 + 3.46410i
 $$n$$: 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.c 2
19.c even 3 1 inner 418.2.e.c 2
19.c even 3 1 7942.2.a.o 1
19.d odd 6 1 7942.2.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.e.c 2 1.a even 1 1 trivial
418.2.e.c 2 19.c even 3 1 inner
7942.2.a.d 1 19.d odd 6 1
7942.2.a.o 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 $$S_{2}^{\mathrm{new}}(418, [\chi])$$:

 $$T_{3}$$ T3 $$T_{5}^{2} - 4T_{5} + 16$$ T5^2 - 4*T5 + 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 4T + 16$$
$7$ $$(T + 4)^{2}$$
$11$ $$(T - 1)^{2}$$
$13$ $$T^{2} + 7T + 49$$
$17$ $$T^{2}$$
$19$ $$T^{2} + T + 19$$
$23$ $$T^{2} + 4T + 16$$
$29$ $$T^{2} - 3T + 9$$
$31$ $$T^{2}$$
$37$ $$(T - 8)^{2}$$
$41$ $$T^{2} - 12T + 144$$
$43$ $$T^{2} + 7T + 49$$
$47$ $$T^{2} - 5T + 25$$
$53$ $$T^{2} + 6T + 36$$
$59$ $$T^{2} + 6T + 36$$
$61$ $$T^{2} - 5T + 25$$
$67$ $$T^{2} - 8T + 64$$
$71$ $$T^{2} - 9T + 81$$
$73$ $$T^{2} + 10T + 100$$
$79$ $$T^{2} - 6T + 36$$
$83$ $$(T + 3)^{2}$$
$89$ $$T^{2} - T + 1$$
$97$ $$T^{2} - 5T + 25$$