# Properties

 Label 414.2.e.a Level $414$ Weight $2$ Character orbit 414.e Analytic conductor $3.306$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$414 = 2 \cdot 3^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 414.e (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.30580664368$$ 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.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{6} + 1) q^{2} + (\zeta_{6} + 1) q^{3} - \zeta_{6} q^{4} - 4 \zeta_{6} q^{5} + ( - \zeta_{6} + 2) q^{6} - q^{8} + 3 \zeta_{6} q^{9} +O(q^{10})$$ q + (-z + 1) * q^2 + (z + 1) * q^3 - z * q^4 - 4*z * q^5 + (-z + 2) * q^6 - q^8 + 3*z * q^9 $$q + ( - \zeta_{6} + 1) q^{2} + (\zeta_{6} + 1) q^{3} - \zeta_{6} q^{4} - 4 \zeta_{6} q^{5} + ( - \zeta_{6} + 2) q^{6} - q^{8} + 3 \zeta_{6} q^{9} - 4 q^{10} + ( - 3 \zeta_{6} + 3) q^{11} + ( - 2 \zeta_{6} + 1) q^{12} - 4 \zeta_{6} q^{13} + ( - 8 \zeta_{6} + 4) q^{15} + (\zeta_{6} - 1) q^{16} + 7 q^{17} + 3 q^{18} - 5 q^{19} + (4 \zeta_{6} - 4) q^{20} - 3 \zeta_{6} q^{22} - \zeta_{6} q^{23} + ( - \zeta_{6} - 1) q^{24} + (11 \zeta_{6} - 11) q^{25} - 4 q^{26} + (6 \zeta_{6} - 3) q^{27} + ( - 4 \zeta_{6} + 4) q^{29} + ( - 4 \zeta_{6} - 4) q^{30} + 2 \zeta_{6} q^{31} + \zeta_{6} q^{32} + ( - 3 \zeta_{6} + 6) q^{33} + ( - 7 \zeta_{6} + 7) q^{34} + ( - 3 \zeta_{6} + 3) q^{36} + 8 q^{37} + (5 \zeta_{6} - 5) q^{38} + ( - 8 \zeta_{6} + 4) q^{39} + 4 \zeta_{6} q^{40} + 7 \zeta_{6} q^{41} + ( - \zeta_{6} + 1) q^{43} - 3 q^{44} + ( - 12 \zeta_{6} + 12) q^{45} - q^{46} + (10 \zeta_{6} - 10) q^{47} + (\zeta_{6} - 2) q^{48} + 7 \zeta_{6} q^{49} + 11 \zeta_{6} q^{50} + (7 \zeta_{6} + 7) q^{51} + (4 \zeta_{6} - 4) q^{52} + 10 q^{53} + (3 \zeta_{6} + 3) q^{54} - 12 q^{55} + ( - 5 \zeta_{6} - 5) q^{57} - 4 \zeta_{6} q^{58} - 13 \zeta_{6} q^{59} + (4 \zeta_{6} - 8) q^{60} + (2 \zeta_{6} - 2) q^{61} + 2 q^{62} + q^{64} + (16 \zeta_{6} - 16) q^{65} + ( - 6 \zeta_{6} + 3) q^{66} + 7 \zeta_{6} q^{67} - 7 \zeta_{6} q^{68} + ( - 2 \zeta_{6} + 1) q^{69} - 3 \zeta_{6} q^{72} - 9 q^{73} + ( - 8 \zeta_{6} + 8) q^{74} + (11 \zeta_{6} - 22) q^{75} + 5 \zeta_{6} q^{76} + ( - 4 \zeta_{6} - 4) q^{78} + (2 \zeta_{6} - 2) q^{79} + 4 q^{80} + (9 \zeta_{6} - 9) q^{81} + 7 q^{82} + (4 \zeta_{6} - 4) q^{83} - 28 \zeta_{6} q^{85} - \zeta_{6} q^{86} + ( - 4 \zeta_{6} + 8) q^{87} + (3 \zeta_{6} - 3) q^{88} + 18 q^{89} - 12 \zeta_{6} q^{90} + (\zeta_{6} - 1) q^{92} + (4 \zeta_{6} - 2) q^{93} + 10 \zeta_{6} q^{94} + 20 \zeta_{6} q^{95} + (2 \zeta_{6} - 1) q^{96} + (7 \zeta_{6} - 7) q^{97} + 7 q^{98} + 9 q^{99} +O(q^{100})$$ q + (-z + 1) * q^2 + (z + 1) * q^3 - z * q^4 - 4*z * q^5 + (-z + 2) * q^6 - q^8 + 3*z * q^9 - 4 * q^10 + (-3*z + 3) * q^11 + (-2*z + 1) * q^12 - 4*z * q^13 + (-8*z + 4) * q^15 + (z - 1) * q^16 + 7 * q^17 + 3 * q^18 - 5 * q^19 + (4*z - 4) * q^20 - 3*z * q^22 - z * q^23 + (-z - 1) * q^24 + (11*z - 11) * q^25 - 4 * q^26 + (6*z - 3) * q^27 + (-4*z + 4) * q^29 + (-4*z - 4) * q^30 + 2*z * q^31 + z * q^32 + (-3*z + 6) * q^33 + (-7*z + 7) * q^34 + (-3*z + 3) * q^36 + 8 * q^37 + (5*z - 5) * q^38 + (-8*z + 4) * q^39 + 4*z * q^40 + 7*z * q^41 + (-z + 1) * q^43 - 3 * q^44 + (-12*z + 12) * q^45 - q^46 + (10*z - 10) * q^47 + (z - 2) * q^48 + 7*z * q^49 + 11*z * q^50 + (7*z + 7) * q^51 + (4*z - 4) * q^52 + 10 * q^53 + (3*z + 3) * q^54 - 12 * q^55 + (-5*z - 5) * q^57 - 4*z * q^58 - 13*z * q^59 + (4*z - 8) * q^60 + (2*z - 2) * q^61 + 2 * q^62 + q^64 + (16*z - 16) * q^65 + (-6*z + 3) * q^66 + 7*z * q^67 - 7*z * q^68 + (-2*z + 1) * q^69 - 3*z * q^72 - 9 * q^73 + (-8*z + 8) * q^74 + (11*z - 22) * q^75 + 5*z * q^76 + (-4*z - 4) * q^78 + (2*z - 2) * q^79 + 4 * q^80 + (9*z - 9) * q^81 + 7 * q^82 + (4*z - 4) * q^83 - 28*z * q^85 - z * q^86 + (-4*z + 8) * q^87 + (3*z - 3) * q^88 + 18 * q^89 - 12*z * q^90 + (z - 1) * q^92 + (4*z - 2) * q^93 + 10*z * q^94 + 20*z * q^95 + (2*z - 1) * q^96 + (7*z - 7) * q^97 + 7 * q^98 + 9 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} + 3 q^{3} - q^{4} - 4 q^{5} + 3 q^{6} - 2 q^{8} + 3 q^{9}+O(q^{10})$$ 2 * q + q^2 + 3 * q^3 - q^4 - 4 * q^5 + 3 * q^6 - 2 * q^8 + 3 * q^9 $$2 q + q^{2} + 3 q^{3} - q^{4} - 4 q^{5} + 3 q^{6} - 2 q^{8} + 3 q^{9} - 8 q^{10} + 3 q^{11} - 4 q^{13} - q^{16} + 14 q^{17} + 6 q^{18} - 10 q^{19} - 4 q^{20} - 3 q^{22} - q^{23} - 3 q^{24} - 11 q^{25} - 8 q^{26} + 4 q^{29} - 12 q^{30} + 2 q^{31} + q^{32} + 9 q^{33} + 7 q^{34} + 3 q^{36} + 16 q^{37} - 5 q^{38} + 4 q^{40} + 7 q^{41} + q^{43} - 6 q^{44} + 12 q^{45} - 2 q^{46} - 10 q^{47} - 3 q^{48} + 7 q^{49} + 11 q^{50} + 21 q^{51} - 4 q^{52} + 20 q^{53} + 9 q^{54} - 24 q^{55} - 15 q^{57} - 4 q^{58} - 13 q^{59} - 12 q^{60} - 2 q^{61} + 4 q^{62} + 2 q^{64} - 16 q^{65} + 7 q^{67} - 7 q^{68} - 3 q^{72} - 18 q^{73} + 8 q^{74} - 33 q^{75} + 5 q^{76} - 12 q^{78} - 2 q^{79} + 8 q^{80} - 9 q^{81} + 14 q^{82} - 4 q^{83} - 28 q^{85} - q^{86} + 12 q^{87} - 3 q^{88} + 36 q^{89} - 12 q^{90} - q^{92} + 10 q^{94} + 20 q^{95} - 7 q^{97} + 14 q^{98} + 18 q^{99}+O(q^{100})$$ 2 * q + q^2 + 3 * q^3 - q^4 - 4 * q^5 + 3 * q^6 - 2 * q^8 + 3 * q^9 - 8 * q^10 + 3 * q^11 - 4 * q^13 - q^16 + 14 * q^17 + 6 * q^18 - 10 * q^19 - 4 * q^20 - 3 * q^22 - q^23 - 3 * q^24 - 11 * q^25 - 8 * q^26 + 4 * q^29 - 12 * q^30 + 2 * q^31 + q^32 + 9 * q^33 + 7 * q^34 + 3 * q^36 + 16 * q^37 - 5 * q^38 + 4 * q^40 + 7 * q^41 + q^43 - 6 * q^44 + 12 * q^45 - 2 * q^46 - 10 * q^47 - 3 * q^48 + 7 * q^49 + 11 * q^50 + 21 * q^51 - 4 * q^52 + 20 * q^53 + 9 * q^54 - 24 * q^55 - 15 * q^57 - 4 * q^58 - 13 * q^59 - 12 * q^60 - 2 * q^61 + 4 * q^62 + 2 * q^64 - 16 * q^65 + 7 * q^67 - 7 * q^68 - 3 * q^72 - 18 * q^73 + 8 * q^74 - 33 * q^75 + 5 * q^76 - 12 * q^78 - 2 * q^79 + 8 * q^80 - 9 * q^81 + 14 * q^82 - 4 * q^83 - 28 * q^85 - q^86 + 12 * q^87 - 3 * q^88 + 36 * q^89 - 12 * q^90 - q^92 + 10 * q^94 + 20 * q^95 - 7 * q^97 + 14 * q^98 + 18 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/414\mathbb{Z}\right)^\times$$.

 $$n$$ $$47$$ $$235$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
139.1
 0.5 − 0.866025i 0.5 + 0.866025i
0.500000 + 0.866025i 1.50000 0.866025i −0.500000 + 0.866025i −2.00000 + 3.46410i 1.50000 + 0.866025i 0 −1.00000 1.50000 2.59808i −4.00000
277.1 0.500000 0.866025i 1.50000 + 0.866025i −0.500000 0.866025i −2.00000 3.46410i 1.50000 0.866025i 0 −1.00000 1.50000 + 2.59808i −4.00000
 $$n$$: 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
9.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 414.2.e.a 2
3.b odd 2 1 1242.2.e.a 2
9.c even 3 1 inner 414.2.e.a 2
9.c even 3 1 3726.2.a.d 1
9.d odd 6 1 1242.2.e.a 2
9.d odd 6 1 3726.2.a.e 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
414.2.e.a 2 1.a even 1 1 trivial
414.2.e.a 2 9.c even 3 1 inner
1242.2.e.a 2 3.b odd 2 1
1242.2.e.a 2 9.d odd 6 1
3726.2.a.d 1 9.c even 3 1
3726.2.a.e 1 9.d odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 4T_{5} + 16$$ acting on $$S_{2}^{\mathrm{new}}(414, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T + 1$$
$3$ $$T^{2} - 3T + 3$$
$5$ $$T^{2} + 4T + 16$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 3T + 9$$
$13$ $$T^{2} + 4T + 16$$
$17$ $$(T - 7)^{2}$$
$19$ $$(T + 5)^{2}$$
$23$ $$T^{2} + T + 1$$
$29$ $$T^{2} - 4T + 16$$
$31$ $$T^{2} - 2T + 4$$
$37$ $$(T - 8)^{2}$$
$41$ $$T^{2} - 7T + 49$$
$43$ $$T^{2} - T + 1$$
$47$ $$T^{2} + 10T + 100$$
$53$ $$(T - 10)^{2}$$
$59$ $$T^{2} + 13T + 169$$
$61$ $$T^{2} + 2T + 4$$
$67$ $$T^{2} - 7T + 49$$
$71$ $$T^{2}$$
$73$ $$(T + 9)^{2}$$
$79$ $$T^{2} + 2T + 4$$
$83$ $$T^{2} + 4T + 16$$
$89$ $$(T - 18)^{2}$$
$97$ $$T^{2} + 7T + 49$$