# Properties

 Label 2394.2.b.h Level $2394$ Weight $2$ Character orbit 2394.b Analytic conductor $19.116$ Analytic rank $0$ Dimension $6$ 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: $$6$$ Coefficient field: 6.0.2803712.1 Defining polynomial: $$x^{6} + 6 x^{4} + 8 x^{2} + 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} -\beta_{1} q^{5} + q^{7} + q^{8} +O(q^{10})$$ $$q + q^{2} + q^{4} -\beta_{1} q^{5} + q^{7} + q^{8} -\beta_{1} q^{10} + ( -\beta_{1} - \beta_{3} ) q^{11} + ( \beta_{3} + \beta_{4} ) q^{13} + q^{14} + q^{16} + ( \beta_{1} + 3 \beta_{3} ) q^{17} + ( -1 - \beta_{1} - \beta_{2} ) q^{19} -\beta_{1} q^{20} + ( -\beta_{1} - \beta_{3} ) q^{22} + ( 2 \beta_{3} - \beta_{4} ) q^{23} + ( -3 + 2 \beta_{5} ) q^{25} + ( \beta_{3} + \beta_{4} ) q^{26} + q^{28} + 4 q^{29} + ( -\beta_{3} + 2 \beta_{4} ) q^{31} + q^{32} + ( \beta_{1} + 3 \beta_{3} ) q^{34} -\beta_{1} q^{35} + ( -\beta_{1} - 3 \beta_{3} + \beta_{4} ) q^{37} + ( -1 - \beta_{1} - \beta_{2} ) q^{38} -\beta_{1} q^{40} + ( -3 + \beta_{2} + \beta_{5} ) q^{41} + ( -1 - \beta_{2} - \beta_{5} ) q^{43} + ( -\beta_{1} - \beta_{3} ) q^{44} + ( 2 \beta_{3} - \beta_{4} ) q^{46} + ( -\beta_{1} + \beta_{4} ) q^{47} + q^{49} + ( -3 + 2 \beta_{5} ) q^{50} + ( \beta_{3} + \beta_{4} ) q^{52} + ( -2 + 2 \beta_{2} ) q^{53} + ( -7 + \beta_{2} + \beta_{5} ) q^{55} + q^{56} + 4 q^{58} + ( 2 + 2 \beta_{5} ) q^{59} + ( -1 + \beta_{2} - \beta_{5} ) q^{61} + ( -\beta_{3} + 2 \beta_{4} ) q^{62} + q^{64} + ( -1 + \beta_{2} - \beta_{5} ) q^{65} + ( -\beta_{1} + 2 \beta_{3} ) q^{67} + ( \beta_{1} + 3 \beta_{3} ) q^{68} -\beta_{1} q^{70} + ( 2 \beta_{2} - 4 \beta_{5} ) q^{71} + ( -3 - \beta_{2} + 3 \beta_{5} ) q^{73} + ( -\beta_{1} - 3 \beta_{3} + \beta_{4} ) q^{74} + ( -1 - \beta_{1} - \beta_{2} ) q^{76} + ( -\beta_{1} - \beta_{3} ) q^{77} + ( 2 \beta_{1} - \beta_{3} - \beta_{4} ) q^{79} -\beta_{1} q^{80} + ( -3 + \beta_{2} + \beta_{5} ) q^{82} + ( \beta_{1} - \beta_{3} - \beta_{4} ) q^{83} + ( 5 - 3 \beta_{2} + \beta_{5} ) q^{85} + ( -1 - \beta_{2} - \beta_{5} ) q^{86} + ( -\beta_{1} - \beta_{3} ) q^{88} + ( 1 + \beta_{2} + \beta_{5} ) q^{89} + ( \beta_{3} + \beta_{4} ) q^{91} + ( 2 \beta_{3} - \beta_{4} ) q^{92} + ( -\beta_{1} + \beta_{4} ) q^{94} + ( -8 - 4 \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{95} + ( -4 \beta_{1} - \beta_{4} ) q^{97} + q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 6 q^{2} + 6 q^{4} + 6 q^{7} + 6 q^{8} + O(q^{10})$$ $$6 q + 6 q^{2} + 6 q^{4} + 6 q^{7} + 6 q^{8} + 6 q^{14} + 6 q^{16} - 6 q^{19} - 14 q^{25} + 6 q^{28} + 24 q^{29} + 6 q^{32} - 6 q^{38} - 16 q^{41} - 8 q^{43} + 6 q^{49} - 14 q^{50} - 12 q^{53} - 40 q^{55} + 6 q^{56} + 24 q^{58} + 16 q^{59} - 8 q^{61} + 6 q^{64} - 8 q^{65} - 8 q^{71} - 12 q^{73} - 6 q^{76} - 16 q^{82} + 32 q^{85} - 8 q^{86} + 8 q^{89} - 44 q^{95} + 6 q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} + 6 x^{4} + 8 x^{2} + 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$-\nu^{5} - 5 \nu^{3} - 2 \nu$$ $$\beta_{2}$$ $$=$$ $$2 \nu^{2} + 4$$ $$\beta_{3}$$ $$=$$ $$\nu^{5} + 5 \nu^{3} + 4 \nu$$ $$\beta_{4}$$ $$=$$ $$\nu^{5} + 7 \nu^{3} + 10 \nu$$ $$\beta_{5}$$ $$=$$ $$2 \nu^{4} + 10 \nu^{2} + 7$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{2} - 4$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{4} - 4 \beta_{3} - 3 \beta_{1}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$\beta_{5} - 5 \beta_{2} + 13$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-5 \beta_{4} + 18 \beta_{3} + 11 \beta_{1}$$$$)/2$$

## 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
 1.20864i 2.05288i − 0.569973i 0.569973i − 2.05288i − 1.20864i
1.00000 0 1.00000 3.83149i 0 1.00000 1.00000 0 3.83149i
1709.2 1.00000 0 1.00000 2.69155i 0 1.00000 1.00000 0 2.69155i
1709.3 1.00000 0 1.00000 0.274268i 0 1.00000 1.00000 0 0.274268i
1709.4 1.00000 0 1.00000 0.274268i 0 1.00000 1.00000 0 0.274268i
1709.5 1.00000 0 1.00000 2.69155i 0 1.00000 1.00000 0 2.69155i
1709.6 1.00000 0 1.00000 3.83149i 0 1.00000 1.00000 0 3.83149i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1709.6 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.h yes 6
3.b odd 2 1 2394.2.b.g 6
19.b odd 2 1 2394.2.b.g 6
57.d even 2 1 inner 2394.2.b.h yes 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2394.2.b.g 6 3.b odd 2 1
2394.2.b.g 6 19.b odd 2 1
2394.2.b.h yes 6 1.a even 1 1 trivial
2394.2.b.h yes 6 57.d even 2 1 inner

## Hecke kernels

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

 $$T_{5}^{6} + 22 T_{5}^{4} + 108 T_{5}^{2} + 8$$ $$T_{29} - 4$$ $$T_{53}^{3} + 6 T_{53}^{2} - 52 T_{53} + 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{6}$$
$3$ $$T^{6}$$
$5$ $$8 + 108 T^{2} + 22 T^{4} + T^{6}$$
$7$ $$( -1 + T )^{6}$$
$11$ $$128 + 128 T^{2} + 24 T^{4} + T^{6}$$
$13$ $$128 + 192 T^{2} + 40 T^{4} + T^{6}$$
$17$ $$128 + 768 T^{2} + 64 T^{4} + T^{6}$$
$19$ $$6859 + 2166 T + 95 T^{2} - 20 T^{3} + 5 T^{4} + 6 T^{5} + T^{6}$$
$23$ $$2888 + 1260 T^{2} + 70 T^{4} + T^{6}$$
$29$ $$( -4 + T )^{6}$$
$31$ $$150152 + 8844 T^{2} + 166 T^{4} + T^{6}$$
$37$ $$200 + 844 T^{2} + 118 T^{4} + T^{6}$$
$41$ $$( -160 - 16 T + 8 T^{2} + T^{3} )^{2}$$
$43$ $$( 32 - 32 T + 4 T^{2} + T^{3} )^{2}$$
$47$ $$2048 + 1024 T^{2} + 64 T^{4} + T^{6}$$
$53$ $$( 8 - 52 T + 6 T^{2} + T^{3} )^{2}$$
$59$ $$( 128 - 32 T - 8 T^{2} + T^{3} )^{2}$$
$61$ $$( -32 - 16 T + 4 T^{2} + T^{3} )^{2}$$
$67$ $$8 + 428 T^{2} + 54 T^{4} + T^{6}$$
$71$ $$( -1472 - 208 T + 4 T^{2} + T^{3} )^{2}$$
$73$ $$( 200 - 100 T + 6 T^{2} + T^{3} )^{2}$$
$79$ $$107648 + 6848 T^{2} + 144 T^{4} + T^{6}$$
$83$ $$7688 + 1420 T^{2} + 70 T^{4} + T^{6}$$
$89$ $$( -32 - 32 T - 4 T^{2} + T^{3} )^{2}$$
$97$ $$182408 + 20204 T^{2} + 374 T^{4} + T^{6}$$