# Properties

 Label 462.2.i.d Level $462$ Weight $2$ Character orbit 462.i Analytic conductor $3.689$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$462 = 2 \cdot 3 \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 462.i (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.68908857338$$ 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.

 $$f(q)$$ $$=$$ $$q + \zeta_{6} q^{2} + ( - \zeta_{6} + 1) q^{3} + (\zeta_{6} - 1) q^{4} + q^{6} + ( - 2 \zeta_{6} + 3) q^{7} - q^{8} - \zeta_{6} q^{9} +O(q^{10})$$ q + z * q^2 + (-z + 1) * q^3 + (z - 1) * q^4 + q^6 + (-2*z + 3) * q^7 - q^8 - z * q^9 $$q + \zeta_{6} q^{2} + ( - \zeta_{6} + 1) q^{3} + (\zeta_{6} - 1) q^{4} + q^{6} + ( - 2 \zeta_{6} + 3) q^{7} - q^{8} - \zeta_{6} q^{9} + ( - \zeta_{6} + 1) q^{11} + \zeta_{6} q^{12} + 4 q^{13} + (\zeta_{6} + 2) q^{14} - \zeta_{6} q^{16} + ( - \zeta_{6} + 1) q^{17} + ( - \zeta_{6} + 1) q^{18} + 3 \zeta_{6} q^{19} + ( - 3 \zeta_{6} + 1) q^{21} + q^{22} + \zeta_{6} q^{23} + (\zeta_{6} - 1) q^{24} + ( - 5 \zeta_{6} + 5) q^{25} + 4 \zeta_{6} q^{26} - q^{27} + (3 \zeta_{6} - 1) q^{28} - q^{29} + (6 \zeta_{6} - 6) q^{31} + ( - \zeta_{6} + 1) q^{32} - \zeta_{6} q^{33} + q^{34} + q^{36} + 3 \zeta_{6} q^{37} + (3 \zeta_{6} - 3) q^{38} + ( - 4 \zeta_{6} + 4) q^{39} - 6 q^{41} + ( - 2 \zeta_{6} + 3) q^{42} + q^{43} + \zeta_{6} q^{44} + (\zeta_{6} - 1) q^{46} + \zeta_{6} q^{47} - q^{48} + ( - 8 \zeta_{6} + 5) q^{49} + 5 q^{50} - \zeta_{6} q^{51} + (4 \zeta_{6} - 4) q^{52} - \zeta_{6} q^{54} + (2 \zeta_{6} - 3) q^{56} + 3 q^{57} - \zeta_{6} q^{58} + (7 \zeta_{6} - 7) q^{59} - 6 \zeta_{6} q^{61} - 6 q^{62} + ( - \zeta_{6} - 2) q^{63} + q^{64} + ( - \zeta_{6} + 1) q^{66} + ( - 4 \zeta_{6} + 4) q^{67} + \zeta_{6} q^{68} + q^{69} - 15 q^{71} + \zeta_{6} q^{72} + (12 \zeta_{6} - 12) q^{73} + (3 \zeta_{6} - 3) q^{74} - 5 \zeta_{6} q^{75} - 3 q^{76} + ( - 3 \zeta_{6} + 1) q^{77} + 4 q^{78} + (\zeta_{6} - 1) q^{81} - 6 \zeta_{6} q^{82} - 16 q^{83} + (\zeta_{6} + 2) q^{84} + \zeta_{6} q^{86} + (\zeta_{6} - 1) q^{87} + (\zeta_{6} - 1) q^{88} + 8 \zeta_{6} q^{89} + ( - 8 \zeta_{6} + 12) q^{91} - q^{92} + 6 \zeta_{6} q^{93} + (\zeta_{6} - 1) q^{94} - \zeta_{6} q^{96} + 7 q^{97} + ( - 3 \zeta_{6} + 8) q^{98} - q^{99} +O(q^{100})$$ q + z * q^2 + (-z + 1) * q^3 + (z - 1) * q^4 + q^6 + (-2*z + 3) * q^7 - q^8 - z * q^9 + (-z + 1) * q^11 + z * q^12 + 4 * q^13 + (z + 2) * q^14 - z * q^16 + (-z + 1) * q^17 + (-z + 1) * q^18 + 3*z * q^19 + (-3*z + 1) * q^21 + q^22 + z * q^23 + (z - 1) * q^24 + (-5*z + 5) * q^25 + 4*z * q^26 - q^27 + (3*z - 1) * q^28 - q^29 + (6*z - 6) * q^31 + (-z + 1) * q^32 - z * q^33 + q^34 + q^36 + 3*z * q^37 + (3*z - 3) * q^38 + (-4*z + 4) * q^39 - 6 * q^41 + (-2*z + 3) * q^42 + q^43 + z * q^44 + (z - 1) * q^46 + z * q^47 - q^48 + (-8*z + 5) * q^49 + 5 * q^50 - z * q^51 + (4*z - 4) * q^52 - z * q^54 + (2*z - 3) * q^56 + 3 * q^57 - z * q^58 + (7*z - 7) * q^59 - 6*z * q^61 - 6 * q^62 + (-z - 2) * q^63 + q^64 + (-z + 1) * q^66 + (-4*z + 4) * q^67 + z * q^68 + q^69 - 15 * q^71 + z * q^72 + (12*z - 12) * q^73 + (3*z - 3) * q^74 - 5*z * q^75 - 3 * q^76 + (-3*z + 1) * q^77 + 4 * q^78 + (z - 1) * q^81 - 6*z * q^82 - 16 * q^83 + (z + 2) * q^84 + z * q^86 + (z - 1) * q^87 + (z - 1) * q^88 + 8*z * q^89 + (-8*z + 12) * q^91 - q^92 + 6*z * q^93 + (z - 1) * q^94 - z * q^96 + 7 * q^97 + (-3*z + 8) * q^98 - q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} + q^{3} - q^{4} + 2 q^{6} + 4 q^{7} - 2 q^{8} - q^{9}+O(q^{10})$$ 2 * q + q^2 + q^3 - q^4 + 2 * q^6 + 4 * q^7 - 2 * q^8 - q^9 $$2 q + q^{2} + q^{3} - q^{4} + 2 q^{6} + 4 q^{7} - 2 q^{8} - q^{9} + q^{11} + q^{12} + 8 q^{13} + 5 q^{14} - q^{16} + q^{17} + q^{18} + 3 q^{19} - q^{21} + 2 q^{22} + q^{23} - q^{24} + 5 q^{25} + 4 q^{26} - 2 q^{27} + q^{28} - 2 q^{29} - 6 q^{31} + q^{32} - q^{33} + 2 q^{34} + 2 q^{36} + 3 q^{37} - 3 q^{38} + 4 q^{39} - 12 q^{41} + 4 q^{42} + 2 q^{43} + q^{44} - q^{46} + q^{47} - 2 q^{48} + 2 q^{49} + 10 q^{50} - q^{51} - 4 q^{52} - q^{54} - 4 q^{56} + 6 q^{57} - q^{58} - 7 q^{59} - 6 q^{61} - 12 q^{62} - 5 q^{63} + 2 q^{64} + q^{66} + 4 q^{67} + q^{68} + 2 q^{69} - 30 q^{71} + q^{72} - 12 q^{73} - 3 q^{74} - 5 q^{75} - 6 q^{76} - q^{77} + 8 q^{78} - q^{81} - 6 q^{82} - 32 q^{83} + 5 q^{84} + q^{86} - q^{87} - q^{88} + 8 q^{89} + 16 q^{91} - 2 q^{92} + 6 q^{93} - q^{94} - q^{96} + 14 q^{97} + 13 q^{98} - 2 q^{99}+O(q^{100})$$ 2 * q + q^2 + q^3 - q^4 + 2 * q^6 + 4 * q^7 - 2 * q^8 - q^9 + q^11 + q^12 + 8 * q^13 + 5 * q^14 - q^16 + q^17 + q^18 + 3 * q^19 - q^21 + 2 * q^22 + q^23 - q^24 + 5 * q^25 + 4 * q^26 - 2 * q^27 + q^28 - 2 * q^29 - 6 * q^31 + q^32 - q^33 + 2 * q^34 + 2 * q^36 + 3 * q^37 - 3 * q^38 + 4 * q^39 - 12 * q^41 + 4 * q^42 + 2 * q^43 + q^44 - q^46 + q^47 - 2 * q^48 + 2 * q^49 + 10 * q^50 - q^51 - 4 * q^52 - q^54 - 4 * q^56 + 6 * q^57 - q^58 - 7 * q^59 - 6 * q^61 - 12 * q^62 - 5 * q^63 + 2 * q^64 + q^66 + 4 * q^67 + q^68 + 2 * q^69 - 30 * q^71 + q^72 - 12 * q^73 - 3 * q^74 - 5 * q^75 - 6 * q^76 - q^77 + 8 * q^78 - q^81 - 6 * q^82 - 32 * q^83 + 5 * q^84 + q^86 - q^87 - q^88 + 8 * q^89 + 16 * q^91 - 2 * q^92 + 6 * q^93 - q^94 - q^96 + 14 * q^97 + 13 * q^98 - 2 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/462\mathbb{Z}\right)^\times$$.

 $$n$$ $$155$$ $$199$$ $$211$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
67.1
 0.5 + 0.866025i 0.5 − 0.866025i
0.500000 + 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i 0 1.00000 2.00000 1.73205i −1.00000 −0.500000 0.866025i 0
331.1 0.500000 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i 0 1.00000 2.00000 + 1.73205i −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
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 462.2.i.d 2
3.b odd 2 1 1386.2.k.f 2
7.c even 3 1 inner 462.2.i.d 2
7.c even 3 1 3234.2.a.c 1
7.d odd 6 1 3234.2.a.j 1
21.g even 6 1 9702.2.a.bq 1
21.h odd 6 1 1386.2.k.f 2
21.h odd 6 1 9702.2.a.bv 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.i.d 2 1.a even 1 1 trivial
462.2.i.d 2 7.c even 3 1 inner
1386.2.k.f 2 3.b odd 2 1
1386.2.k.f 2 21.h odd 6 1
3234.2.a.c 1 7.c even 3 1
3234.2.a.j 1 7.d odd 6 1
9702.2.a.bq 1 21.g even 6 1
9702.2.a.bv 1 21.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(462, [\chi])$$:

 $$T_{5}$$ T5 $$T_{13} - 4$$ T13 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T + 1$$
$3$ $$T^{2} - T + 1$$
$5$ $$T^{2}$$
$7$ $$T^{2} - 4T + 7$$
$11$ $$T^{2} - T + 1$$
$13$ $$(T - 4)^{2}$$
$17$ $$T^{2} - T + 1$$
$19$ $$T^{2} - 3T + 9$$
$23$ $$T^{2} - T + 1$$
$29$ $$(T + 1)^{2}$$
$31$ $$T^{2} + 6T + 36$$
$37$ $$T^{2} - 3T + 9$$
$41$ $$(T + 6)^{2}$$
$43$ $$(T - 1)^{2}$$
$47$ $$T^{2} - T + 1$$
$53$ $$T^{2}$$
$59$ $$T^{2} + 7T + 49$$
$61$ $$T^{2} + 6T + 36$$
$67$ $$T^{2} - 4T + 16$$
$71$ $$(T + 15)^{2}$$
$73$ $$T^{2} + 12T + 144$$
$79$ $$T^{2}$$
$83$ $$(T + 16)^{2}$$
$89$ $$T^{2} - 8T + 64$$
$97$ $$(T - 7)^{2}$$