# Properties

 Label 2394.2.o.b 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: $$x^{2} - x + 1$$ 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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \zeta_{6} ) q^{2} -\zeta_{6} q^{4} + ( -3 + 3 \zeta_{6} ) q^{5} + q^{7} + q^{8} +O(q^{10})$$ $$q + ( -1 + \zeta_{6} ) q^{2} -\zeta_{6} q^{4} + ( -3 + 3 \zeta_{6} ) q^{5} + q^{7} + q^{8} -3 \zeta_{6} q^{10} + 6 q^{11} -5 \zeta_{6} q^{13} + ( -1 + \zeta_{6} ) q^{14} + ( -1 + \zeta_{6} ) q^{16} + ( -6 + 6 \zeta_{6} ) q^{17} + ( -3 + 5 \zeta_{6} ) q^{19} + 3 q^{20} + ( -6 + 6 \zeta_{6} ) q^{22} -3 \zeta_{6} q^{23} -4 \zeta_{6} q^{25} + 5 q^{26} -\zeta_{6} q^{28} + 6 \zeta_{6} q^{29} -4 q^{31} -\zeta_{6} q^{32} -6 \zeta_{6} q^{34} + ( -3 + 3 \zeta_{6} ) q^{35} + 8 q^{37} + ( -2 - 3 \zeta_{6} ) q^{38} + ( -3 + 3 \zeta_{6} ) q^{40} + ( 6 - 6 \zeta_{6} ) q^{41} + ( 4 - 4 \zeta_{6} ) q^{43} -6 \zeta_{6} q^{44} + 3 q^{46} + 6 \zeta_{6} q^{47} + q^{49} + 4 q^{50} + ( -5 + 5 \zeta_{6} ) q^{52} + 12 \zeta_{6} q^{53} + ( -18 + 18 \zeta_{6} ) q^{55} + q^{56} -6 q^{58} + ( -9 + 9 \zeta_{6} ) q^{59} + 13 \zeta_{6} q^{61} + ( 4 - 4 \zeta_{6} ) q^{62} + q^{64} + 15 q^{65} -14 \zeta_{6} q^{67} + 6 q^{68} -3 \zeta_{6} q^{70} + ( -3 + 3 \zeta_{6} ) q^{71} + ( 4 - 4 \zeta_{6} ) q^{73} + ( -8 + 8 \zeta_{6} ) q^{74} + ( 5 - 2 \zeta_{6} ) q^{76} + 6 q^{77} + ( -8 + 8 \zeta_{6} ) q^{79} -3 \zeta_{6} q^{80} + 6 \zeta_{6} q^{82} -15 q^{83} -18 \zeta_{6} q^{85} + 4 \zeta_{6} q^{86} + 6 q^{88} -6 \zeta_{6} q^{89} -5 \zeta_{6} q^{91} + ( -3 + 3 \zeta_{6} ) q^{92} -6 q^{94} + ( -6 - 9 \zeta_{6} ) q^{95} + ( -8 + 8 \zeta_{6} ) q^{97} + ( -1 + \zeta_{6} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} - q^{4} - 3 q^{5} + 2 q^{7} + 2 q^{8} + O(q^{10})$$ $$2 q - q^{2} - q^{4} - 3 q^{5} + 2 q^{7} + 2 q^{8} - 3 q^{10} + 12 q^{11} - 5 q^{13} - q^{14} - q^{16} - 6 q^{17} - q^{19} + 6 q^{20} - 6 q^{22} - 3 q^{23} - 4 q^{25} + 10 q^{26} - q^{28} + 6 q^{29} - 8 q^{31} - q^{32} - 6 q^{34} - 3 q^{35} + 16 q^{37} - 7 q^{38} - 3 q^{40} + 6 q^{41} + 4 q^{43} - 6 q^{44} + 6 q^{46} + 6 q^{47} + 2 q^{49} + 8 q^{50} - 5 q^{52} + 12 q^{53} - 18 q^{55} + 2 q^{56} - 12 q^{58} - 9 q^{59} + 13 q^{61} + 4 q^{62} + 2 q^{64} + 30 q^{65} - 14 q^{67} + 12 q^{68} - 3 q^{70} - 3 q^{71} + 4 q^{73} - 8 q^{74} + 8 q^{76} + 12 q^{77} - 8 q^{79} - 3 q^{80} + 6 q^{82} - 30 q^{83} - 18 q^{85} + 4 q^{86} + 12 q^{88} - 6 q^{89} - 5 q^{91} - 3 q^{92} - 12 q^{94} - 21 q^{95} - 8 q^{97} - q^{98} + O(q^{100})$$

## 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.b 2
3.b odd 2 1 798.2.k.h 2
19.c even 3 1 inner 2394.2.o.b 2
57.h odd 6 1 798.2.k.h 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
798.2.k.h 2 3.b odd 2 1
798.2.k.h 2 57.h odd 6 1
2394.2.o.b 2 1.a even 1 1 trivial
2394.2.o.b 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 $$S_{2}^{\mathrm{new}}(2394, [\chi])$$:

 $$T_{5}^{2} + 3 T_{5} + 9$$ $$T_{11} - 6$$ $$T_{13}^{2} + 5 T_{13} + 25$$ $$T_{17}^{2} + 6 T_{17} + 36$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$9 + 3 T + T^{2}$$
$7$ $$( -1 + T )^{2}$$
$11$ $$( -6 + T )^{2}$$
$13$ $$25 + 5 T + T^{2}$$
$17$ $$36 + 6 T + T^{2}$$
$19$ $$19 + T + T^{2}$$
$23$ $$9 + 3 T + T^{2}$$
$29$ $$36 - 6 T + T^{2}$$
$31$ $$( 4 + T )^{2}$$
$37$ $$( -8 + T )^{2}$$
$41$ $$36 - 6 T + T^{2}$$
$43$ $$16 - 4 T + T^{2}$$
$47$ $$36 - 6 T + T^{2}$$
$53$ $$144 - 12 T + T^{2}$$
$59$ $$81 + 9 T + T^{2}$$
$61$ $$169 - 13 T + T^{2}$$
$67$ $$196 + 14 T + T^{2}$$
$71$ $$9 + 3 T + T^{2}$$
$73$ $$16 - 4 T + T^{2}$$
$79$ $$64 + 8 T + T^{2}$$
$83$ $$( 15 + T )^{2}$$
$89$ $$36 + 6 T + T^{2}$$
$97$ $$64 + 8 T + T^{2}$$