# Properties

 Label 2394.2.o.a Level $2394$ Weight $2$ Character orbit 2394.o Analytic conductor $19.116$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2394 = 2 \cdot 3^{2} \cdot 7 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2394.o (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$19.1161862439$$ 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: no (minimal twist has level 798) 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.

 $$f(q)$$ $$=$$ $$q + (\zeta_{6} - 1) q^{2} - \zeta_{6} q^{4} + (3 \zeta_{6} - 3) q^{5} - q^{7} + q^{8} +O(q^{10})$$ q + (z - 1) * q^2 - z * q^4 + (3*z - 3) * q^5 - q^7 + q^8 $$q + (\zeta_{6} - 1) q^{2} - \zeta_{6} q^{4} + (3 \zeta_{6} - 3) q^{5} - q^{7} + q^{8} - 3 \zeta_{6} q^{10} - 4 q^{11} - 3 \zeta_{6} q^{13} + ( - \zeta_{6} + 1) q^{14} + (\zeta_{6} - 1) q^{16} + (2 \zeta_{6} - 2) q^{17} + (5 \zeta_{6} - 3) q^{19} + 3 q^{20} + ( - 4 \zeta_{6} + 4) q^{22} + \zeta_{6} q^{23} - 4 \zeta_{6} q^{25} + 3 q^{26} + \zeta_{6} q^{28} - 6 q^{31} - \zeta_{6} q^{32} - 2 \zeta_{6} q^{34} + ( - 3 \zeta_{6} + 3) q^{35} + 4 q^{37} + ( - 3 \zeta_{6} - 2) q^{38} + (3 \zeta_{6} - 3) q^{40} + ( - 4 \zeta_{6} + 4) q^{41} + ( - 6 \zeta_{6} + 6) q^{43} + 4 \zeta_{6} q^{44} - q^{46} - 2 \zeta_{6} q^{47} + q^{49} + 4 q^{50} + (3 \zeta_{6} - 3) q^{52} - 4 \zeta_{6} q^{53} + ( - 12 \zeta_{6} + 12) q^{55} - q^{56} + ( - 13 \zeta_{6} + 13) q^{59} - 5 \zeta_{6} q^{61} + ( - 6 \zeta_{6} + 6) q^{62} + q^{64} + 9 q^{65} + 12 \zeta_{6} q^{67} + 2 q^{68} + 3 \zeta_{6} q^{70} + (7 \zeta_{6} - 7) q^{71} + ( - 2 \zeta_{6} + 2) q^{73} + (4 \zeta_{6} - 4) q^{74} + ( - 2 \zeta_{6} + 5) q^{76} + 4 q^{77} + ( - 8 \zeta_{6} + 8) q^{79} - 3 \zeta_{6} q^{80} + 4 \zeta_{6} q^{82} - 17 q^{83} - 6 \zeta_{6} q^{85} + 6 \zeta_{6} q^{86} - 4 q^{88} + 4 \zeta_{6} q^{89} + 3 \zeta_{6} q^{91} + ( - \zeta_{6} + 1) q^{92} + 2 q^{94} + ( - 9 \zeta_{6} - 6) q^{95} + ( - 10 \zeta_{6} + 10) q^{97} + (\zeta_{6} - 1) q^{98} +O(q^{100})$$ q + (z - 1) * q^2 - z * q^4 + (3*z - 3) * q^5 - q^7 + q^8 - 3*z * q^10 - 4 * q^11 - 3*z * q^13 + (-z + 1) * q^14 + (z - 1) * q^16 + (2*z - 2) * q^17 + (5*z - 3) * q^19 + 3 * q^20 + (-4*z + 4) * q^22 + z * q^23 - 4*z * q^25 + 3 * q^26 + z * q^28 - 6 * q^31 - z * q^32 - 2*z * q^34 + (-3*z + 3) * q^35 + 4 * q^37 + (-3*z - 2) * q^38 + (3*z - 3) * q^40 + (-4*z + 4) * q^41 + (-6*z + 6) * q^43 + 4*z * q^44 - q^46 - 2*z * q^47 + q^49 + 4 * q^50 + (3*z - 3) * q^52 - 4*z * q^53 + (-12*z + 12) * q^55 - q^56 + (-13*z + 13) * q^59 - 5*z * q^61 + (-6*z + 6) * q^62 + q^64 + 9 * q^65 + 12*z * q^67 + 2 * q^68 + 3*z * q^70 + (7*z - 7) * q^71 + (-2*z + 2) * q^73 + (4*z - 4) * q^74 + (-2*z + 5) * q^76 + 4 * q^77 + (-8*z + 8) * q^79 - 3*z * q^80 + 4*z * q^82 - 17 * q^83 - 6*z * q^85 + 6*z * q^86 - 4 * q^88 + 4*z * q^89 + 3*z * q^91 + (-z + 1) * q^92 + 2 * q^94 + (-9*z - 6) * q^95 + (-10*z + 10) * q^97 + (z - 1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} - q^{4} - 3 q^{5} - 2 q^{7} + 2 q^{8}+O(q^{10})$$ 2 * q - q^2 - q^4 - 3 * q^5 - 2 * q^7 + 2 * q^8 $$2 q - q^{2} - q^{4} - 3 q^{5} - 2 q^{7} + 2 q^{8} - 3 q^{10} - 8 q^{11} - 3 q^{13} + q^{14} - q^{16} - 2 q^{17} - q^{19} + 6 q^{20} + 4 q^{22} + q^{23} - 4 q^{25} + 6 q^{26} + q^{28} - 12 q^{31} - q^{32} - 2 q^{34} + 3 q^{35} + 8 q^{37} - 7 q^{38} - 3 q^{40} + 4 q^{41} + 6 q^{43} + 4 q^{44} - 2 q^{46} - 2 q^{47} + 2 q^{49} + 8 q^{50} - 3 q^{52} - 4 q^{53} + 12 q^{55} - 2 q^{56} + 13 q^{59} - 5 q^{61} + 6 q^{62} + 2 q^{64} + 18 q^{65} + 12 q^{67} + 4 q^{68} + 3 q^{70} - 7 q^{71} + 2 q^{73} - 4 q^{74} + 8 q^{76} + 8 q^{77} + 8 q^{79} - 3 q^{80} + 4 q^{82} - 34 q^{83} - 6 q^{85} + 6 q^{86} - 8 q^{88} + 4 q^{89} + 3 q^{91} + q^{92} + 4 q^{94} - 21 q^{95} + 10 q^{97} - q^{98}+O(q^{100})$$ 2 * q - q^2 - q^4 - 3 * q^5 - 2 * q^7 + 2 * q^8 - 3 * q^10 - 8 * q^11 - 3 * q^13 + q^14 - q^16 - 2 * q^17 - q^19 + 6 * q^20 + 4 * q^22 + q^23 - 4 * q^25 + 6 * q^26 + q^28 - 12 * q^31 - q^32 - 2 * q^34 + 3 * q^35 + 8 * q^37 - 7 * q^38 - 3 * q^40 + 4 * q^41 + 6 * q^43 + 4 * q^44 - 2 * q^46 - 2 * q^47 + 2 * q^49 + 8 * q^50 - 3 * q^52 - 4 * q^53 + 12 * q^55 - 2 * q^56 + 13 * q^59 - 5 * q^61 + 6 * q^62 + 2 * q^64 + 18 * q^65 + 12 * q^67 + 4 * q^68 + 3 * q^70 - 7 * q^71 + 2 * q^73 - 4 * q^74 + 8 * q^76 + 8 * q^77 + 8 * q^79 - 3 * q^80 + 4 * q^82 - 34 * q^83 - 6 * q^85 + 6 * q^86 - 8 * q^88 + 4 * q^89 + 3 * q^91 + q^92 + 4 * q^94 - 21 * q^95 + 10 * q^97 - q^98

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2394\mathbb{Z}\right)^\times$$.

 $$n$$ $$533$$ $$1009$$ $$1711$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
505.1
 0.5 + 0.866025i 0.5 − 0.866025i
−0.500000 + 0.866025i 0 −0.500000 0.866025i −1.50000 + 2.59808i 0 −1.00000 1.00000 0 −1.50000 2.59808i
1261.1 −0.500000 0.866025i 0 −0.500000 + 0.866025i −1.50000 2.59808i 0 −1.00000 1.00000 0 −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 2394.2.o.a 2
3.b odd 2 1 798.2.k.i 2
19.c even 3 1 inner 2394.2.o.a 2
57.h odd 6 1 798.2.k.i 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
798.2.k.i 2 3.b odd 2 1
798.2.k.i 2 57.h odd 6 1
2394.2.o.a 2 1.a even 1 1 trivial
2394.2.o.a 2 19.c even 3 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2394, [\chi])$$:

 $$T_{5}^{2} + 3T_{5} + 9$$ T5^2 + 3*T5 + 9 $$T_{11} + 4$$ T11 + 4 $$T_{13}^{2} + 3T_{13} + 9$$ T13^2 + 3*T13 + 9 $$T_{17}^{2} + 2T_{17} + 4$$ T17^2 + 2*T17 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 3T + 9$$
$7$ $$(T + 1)^{2}$$
$11$ $$(T + 4)^{2}$$
$13$ $$T^{2} + 3T + 9$$
$17$ $$T^{2} + 2T + 4$$
$19$ $$T^{2} + T + 19$$
$23$ $$T^{2} - T + 1$$
$29$ $$T^{2}$$
$31$ $$(T + 6)^{2}$$
$37$ $$(T - 4)^{2}$$
$41$ $$T^{2} - 4T + 16$$
$43$ $$T^{2} - 6T + 36$$
$47$ $$T^{2} + 2T + 4$$
$53$ $$T^{2} + 4T + 16$$
$59$ $$T^{2} - 13T + 169$$
$61$ $$T^{2} + 5T + 25$$
$67$ $$T^{2} - 12T + 144$$
$71$ $$T^{2} + 7T + 49$$
$73$ $$T^{2} - 2T + 4$$
$79$ $$T^{2} - 8T + 64$$
$83$ $$(T + 17)^{2}$$
$89$ $$T^{2} - 4T + 16$$
$97$ $$T^{2} - 10T + 100$$