# Properties

 Label 2394.2.b.b Level $2394$ Weight $2$ Character orbit 2394.b 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.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$19.1161862439$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ Defining polynomial: $$x^{2} + 2$$ x^2 + 2 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{-2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} + 2 \beta q^{5} + q^{7} - q^{8}+O(q^{10})$$ q - q^2 + q^4 + 2*b * q^5 + q^7 - q^8 $$q - q^{2} + q^{4} + 2 \beta q^{5} + q^{7} - q^{8} - 2 \beta q^{10} - 4 \beta q^{11} + 3 \beta q^{13} - q^{14} + q^{16} - 4 \beta q^{17} + (3 \beta + 1) q^{19} + 2 \beta q^{20} + 4 \beta q^{22} - 4 \beta q^{23} - 3 q^{25} - 3 \beta q^{26} + q^{28} + 6 q^{29} - 6 \beta q^{31} - q^{32} + 4 \beta q^{34} + 2 \beta q^{35} + ( - 3 \beta - 1) q^{38} - 2 \beta q^{40} - 4 q^{43} - 4 \beta q^{44} + 4 \beta q^{46} - 7 \beta q^{47} + q^{49} + 3 q^{50} + 3 \beta q^{52} + 6 q^{53} + 16 q^{55} - q^{56} - 6 q^{58} - 12 q^{59} + 2 q^{61} + 6 \beta q^{62} + q^{64} - 12 q^{65} - 9 \beta q^{67} - 4 \beta q^{68} - 2 \beta q^{70} - 6 q^{71} + 14 q^{73} + (3 \beta + 1) q^{76} - 4 \beta q^{77} - 9 \beta q^{79} + 2 \beta q^{80} + 11 \beta q^{83} + 16 q^{85} + 4 q^{86} + 4 \beta q^{88} + 6 q^{89} + 3 \beta q^{91} - 4 \beta q^{92} + 7 \beta q^{94} + (2 \beta - 12) q^{95} + 3 \beta q^{97} - q^{98} +O(q^{100})$$ q - q^2 + q^4 + 2*b * q^5 + q^7 - q^8 - 2*b * q^10 - 4*b * q^11 + 3*b * q^13 - q^14 + q^16 - 4*b * q^17 + (3*b + 1) * q^19 + 2*b * q^20 + 4*b * q^22 - 4*b * q^23 - 3 * q^25 - 3*b * q^26 + q^28 + 6 * q^29 - 6*b * q^31 - q^32 + 4*b * q^34 + 2*b * q^35 + (-3*b - 1) * q^38 - 2*b * q^40 - 4 * q^43 - 4*b * q^44 + 4*b * q^46 - 7*b * q^47 + q^49 + 3 * q^50 + 3*b * q^52 + 6 * q^53 + 16 * q^55 - q^56 - 6 * q^58 - 12 * q^59 + 2 * q^61 + 6*b * q^62 + q^64 - 12 * q^65 - 9*b * q^67 - 4*b * q^68 - 2*b * q^70 - 6 * q^71 + 14 * q^73 + (3*b + 1) * q^76 - 4*b * q^77 - 9*b * q^79 + 2*b * q^80 + 11*b * q^83 + 16 * q^85 + 4 * q^86 + 4*b * q^88 + 6 * q^89 + 3*b * q^91 - 4*b * q^92 + 7*b * q^94 + (2*b - 12) * q^95 + 3*b * q^97 - q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 2 q^{4} + 2 q^{7} - 2 q^{8}+O(q^{10})$$ 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^7 - 2 * q^8 $$2 q - 2 q^{2} + 2 q^{4} + 2 q^{7} - 2 q^{8} - 2 q^{14} + 2 q^{16} + 2 q^{19} - 6 q^{25} + 2 q^{28} + 12 q^{29} - 2 q^{32} - 2 q^{38} - 8 q^{43} + 2 q^{49} + 6 q^{50} + 12 q^{53} + 32 q^{55} - 2 q^{56} - 12 q^{58} - 24 q^{59} + 4 q^{61} + 2 q^{64} - 24 q^{65} - 12 q^{71} + 28 q^{73} + 2 q^{76} + 32 q^{85} + 8 q^{86} + 12 q^{89} - 24 q^{95} - 2 q^{98}+O(q^{100})$$ 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^7 - 2 * q^8 - 2 * q^14 + 2 * q^16 + 2 * q^19 - 6 * q^25 + 2 * q^28 + 12 * q^29 - 2 * q^32 - 2 * q^38 - 8 * q^43 + 2 * q^49 + 6 * q^50 + 12 * q^53 + 32 * q^55 - 2 * q^56 - 12 * q^58 - 24 * q^59 + 4 * q^61 + 2 * q^64 - 24 * q^65 - 12 * q^71 + 28 * q^73 + 2 * q^76 + 32 * q^85 + 8 * q^86 + 12 * q^89 - 24 * q^95 - 2 * q^98

## Character values

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

 $$n$$ $$533$$ $$1009$$ $$1711$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1709.1
 − 1.41421i 1.41421i
−1.00000 0 1.00000 2.82843i 0 1.00000 −1.00000 0 2.82843i
1709.2 −1.00000 0 1.00000 2.82843i 0 1.00000 −1.00000 0 2.82843i
 $$n$$: 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
57.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2394.2.b.b 2
3.b odd 2 1 2394.2.b.e yes 2
19.b odd 2 1 2394.2.b.e yes 2
57.d even 2 1 inner 2394.2.b.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2394.2.b.b 2 1.a even 1 1 trivial
2394.2.b.b 2 57.d even 2 1 inner
2394.2.b.e yes 2 3.b odd 2 1
2394.2.b.e yes 2 19.b odd 2 1

## Hecke kernels

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

 $$T_{5}^{2} + 8$$ T5^2 + 8 $$T_{29} - 6$$ T29 - 6 $$T_{53} - 6$$ T53 - 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 8$$
$7$ $$(T - 1)^{2}$$
$11$ $$T^{2} + 32$$
$13$ $$T^{2} + 18$$
$17$ $$T^{2} + 32$$
$19$ $$T^{2} - 2T + 19$$
$23$ $$T^{2} + 32$$
$29$ $$(T - 6)^{2}$$
$31$ $$T^{2} + 72$$
$37$ $$T^{2}$$
$41$ $$T^{2}$$
$43$ $$(T + 4)^{2}$$
$47$ $$T^{2} + 98$$
$53$ $$(T - 6)^{2}$$
$59$ $$(T + 12)^{2}$$
$61$ $$(T - 2)^{2}$$
$67$ $$T^{2} + 162$$
$71$ $$(T + 6)^{2}$$
$73$ $$(T - 14)^{2}$$
$79$ $$T^{2} + 162$$
$83$ $$T^{2} + 242$$
$89$ $$(T - 6)^{2}$$
$97$ $$T^{2} + 18$$