# Properties

 Label 2394.2.b.i Level $2394$ Weight $2$ Character orbit 2394.b Analytic conductor $19.116$ Analytic rank $0$ Dimension $8$ 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: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ Defining polynomial: $$x^{8} + 10 x^{6} + 30 x^{4} + 24 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{5}\cdot 3$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} + \beta_{3} q^{5} - q^{7} - q^{8} +O(q^{10})$$ $$q - q^{2} + q^{4} + \beta_{3} q^{5} - q^{7} - q^{8} -\beta_{3} q^{10} -\beta_{5} q^{11} + ( \beta_{5} + \beta_{6} - \beta_{7} ) q^{13} + q^{14} + q^{16} -\beta_{5} q^{17} + ( 1 + \beta_{4} ) q^{19} + \beta_{3} q^{20} + \beta_{5} q^{22} + ( -\beta_{2} + \beta_{5} ) q^{23} + ( -3 + \beta_{4} + \beta_{7} ) q^{25} + ( -\beta_{5} - \beta_{6} + \beta_{7} ) q^{26} - q^{28} + ( 2 - \beta_{4} - \beta_{7} ) q^{29} + ( \beta_{3} + \beta_{5} ) q^{31} - q^{32} + \beta_{5} q^{34} -\beta_{3} q^{35} + ( \beta_{2} - 2 \beta_{3} ) q^{37} + ( -1 - \beta_{4} ) q^{38} -\beta_{3} q^{40} + ( -5 + \beta_{1} + 2 \beta_{4} - 2 \beta_{6} ) q^{41} + ( -5 - \beta_{1} + \beta_{4} - \beta_{6} ) q^{43} -\beta_{5} q^{44} + ( \beta_{2} - \beta_{5} ) q^{46} + ( 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{47} + q^{49} + ( 3 - \beta_{4} - \beta_{7} ) q^{50} + ( \beta_{5} + \beta_{6} - \beta_{7} ) q^{52} + ( -6 + 2 \beta_{1} + \beta_{4} + \beta_{7} ) q^{53} + ( -1 - 3 \beta_{1} - \beta_{6} - \beta_{7} ) q^{55} + q^{56} + ( -2 + \beta_{4} + \beta_{7} ) q^{58} + ( 1 - \beta_{1} - 2 \beta_{4} - 2 \beta_{7} ) q^{61} + ( -\beta_{3} - \beta_{5} ) q^{62} + q^{64} + ( 1 - \beta_{1} - 4 \beta_{4} + 3 \beta_{6} - \beta_{7} ) q^{65} + ( \beta_{2} + \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{67} -\beta_{5} q^{68} + \beta_{3} q^{70} + ( 2 + 2 \beta_{1} ) q^{71} + ( -1 + \beta_{1} + \beta_{4} + \beta_{6} + 2 \beta_{7} ) q^{73} + ( -\beta_{2} + 2 \beta_{3} ) q^{74} + ( 1 + \beta_{4} ) q^{76} + \beta_{5} q^{77} + ( 2 \beta_{2} - 2 \beta_{3} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{79} + \beta_{3} q^{80} + ( 5 - \beta_{1} - 2 \beta_{4} + 2 \beta_{6} ) q^{82} + ( 2 \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{83} + ( -1 - 3 \beta_{1} - \beta_{6} - \beta_{7} ) q^{85} + ( 5 + \beta_{1} - \beta_{4} + \beta_{6} ) q^{86} + \beta_{5} q^{88} + ( 3 - 3 \beta_{1} - 2 \beta_{4} + 2 \beta_{6} ) q^{89} + ( -\beta_{5} - \beta_{6} + \beta_{7} ) q^{91} + ( -\beta_{2} + \beta_{5} ) q^{92} + ( -2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{94} + ( -2 \beta_{1} - \beta_{2} - 2 \beta_{4} + 3 \beta_{5} + 3 \beta_{6} - 3 \beta_{7} ) q^{95} + ( -\beta_{3} + 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{97} - q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 8 q^{2} + 8 q^{4} - 8 q^{7} - 8 q^{8} + O(q^{10})$$ $$8 q - 8 q^{2} + 8 q^{4} - 8 q^{7} - 8 q^{8} + 8 q^{14} + 8 q^{16} + 12 q^{19} - 24 q^{25} - 8 q^{28} + 16 q^{29} - 8 q^{32} - 12 q^{38} - 24 q^{41} - 32 q^{43} + 8 q^{49} + 24 q^{50} - 48 q^{53} + 8 q^{56} - 16 q^{58} + 8 q^{61} + 8 q^{64} - 16 q^{65} + 16 q^{71} - 16 q^{73} + 12 q^{76} + 24 q^{82} + 32 q^{86} + 8 q^{89} - 8 q^{95} - 8 q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 10 x^{6} + 30 x^{4} + 24 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu^{4} + 10 \nu^{2} + 5$$ $$\beta_{2}$$ $$=$$ $$-3 \nu^{5} - 18 \nu^{3} - 21 \nu$$ $$\beta_{3}$$ $$=$$ $$\nu^{7} + 8 \nu^{5} + 19 \nu^{3} + 14 \nu$$ $$\beta_{4}$$ $$=$$ $$\nu^{7} + \nu^{6} + 10 \nu^{5} + 5 \nu^{4} + 29 \nu^{3} + 3 \nu^{2} + 20 \nu + 1$$ $$\beta_{5}$$ $$=$$ $$-\nu^{7} - 11 \nu^{5} - 35 \nu^{3} - 25 \nu$$ $$\beta_{6}$$ $$=$$ $$\nu^{7} - \nu^{6} + 10 \nu^{5} - 7 \nu^{4} + 29 \nu^{3} - 11 \nu^{2} + 20 \nu - 1$$ $$\beta_{7}$$ $$=$$ $$-\nu^{7} - \nu^{6} - 10 \nu^{5} - 7 \nu^{4} - 29 \nu^{3} - 11 \nu^{2} - 20 \nu - 1$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-3 \beta_{7} + 3 \beta_{6} + 6 \beta_{5} - 2 \beta_{2}$$$$)/12$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{7} + \beta_{4} + \beta_{1} - 5$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$15 \beta_{7} - 15 \beta_{6} - 24 \beta_{5} + 6 \beta_{3} + 4 \beta_{2}$$$$)/12$$ $$\nu^{4}$$ $$=$$ $$($$$$-5 \beta_{7} - 5 \beta_{4} - 4 \beta_{1} + 20$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-69 \beta_{7} + 69 \beta_{6} + 102 \beta_{5} - 36 \beta_{3} - 14 \beta_{2}$$$$)/12$$ $$\nu^{6}$$ $$=$$ $$($$$$23 \beta_{7} - \beta_{6} + 24 \beta_{4} + 17 \beta_{1} - 87$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$309 \beta_{7} - 309 \beta_{6} - 444 \beta_{5} + 186 \beta_{3} + 64 \beta_{2}$$$$)/12$$

## 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
 2.15177i − 2.03530i − 0.209899i − 1.08784i 1.08784i 0.209899i 2.03530i − 2.15177i
−1.00000 0 1.00000 3.71953i 0 −1.00000 −1.00000 0 3.71953i
1709.2 −1.00000 0 1.00000 3.02986i 0 −1.00000 −1.00000 0 3.02986i
1709.3 −1.00000 0 1.00000 2.76612i 0 −1.00000 −1.00000 0 2.76612i
1709.4 −1.00000 0 1.00000 1.15484i 0 −1.00000 −1.00000 0 1.15484i
1709.5 −1.00000 0 1.00000 1.15484i 0 −1.00000 −1.00000 0 1.15484i
1709.6 −1.00000 0 1.00000 2.76612i 0 −1.00000 −1.00000 0 2.76612i
1709.7 −1.00000 0 1.00000 3.02986i 0 −1.00000 −1.00000 0 3.02986i
1709.8 −1.00000 0 1.00000 3.71953i 0 −1.00000 −1.00000 0 3.71953i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1709.8 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.i 8
3.b odd 2 1 2394.2.b.j yes 8
19.b odd 2 1 2394.2.b.j yes 8
57.d even 2 1 inner 2394.2.b.i 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2394.2.b.i 8 1.a even 1 1 trivial
2394.2.b.i 8 57.d even 2 1 inner
2394.2.b.j yes 8 3.b odd 2 1
2394.2.b.j yes 8 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 $$S_{2}^{\mathrm{new}}(2394, [\chi])$$:

 $$T_{5}^{8} + 32 T_{5}^{6} + 344 T_{5}^{4} + 1376 T_{5}^{2} + 1296$$ $$T_{29}^{4} - 8 T_{29}^{3} - 16 T_{29}^{2} + 160 T_{29} - 192$$ $$T_{53}^{4} + 24 T_{53}^{3} + 152 T_{53}^{2} - 128 T_{53} - 2544$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{8}$$
$3$ $$T^{8}$$
$5$ $$1296 + 1376 T^{2} + 344 T^{4} + 32 T^{6} + T^{8}$$
$7$ $$( 1 + T )^{8}$$
$11$ $$5184 + 5600 T^{2} + 992 T^{4} + 56 T^{6} + T^{8}$$
$13$ $$14400 + 10208 T^{2} + 1536 T^{4} + 72 T^{6} + T^{8}$$
$17$ $$5184 + 5600 T^{2} + 992 T^{4} + 56 T^{6} + T^{8}$$
$19$ $$130321 - 82308 T + 15884 T^{2} + 1292 T^{3} - 986 T^{4} + 68 T^{5} + 44 T^{6} - 12 T^{7} + T^{8}$$
$23$ $$144 + 992 T^{2} + 1496 T^{4} + 80 T^{6} + T^{8}$$
$29$ $$( -192 + 160 T - 16 T^{2} - 8 T^{3} + T^{4} )^{2}$$
$31$ $$32400 + 14048 T^{2} + 1912 T^{4} + 88 T^{6} + T^{8}$$
$37$ $$138384 + 60128 T^{2} + 6008 T^{4} + 152 T^{6} + T^{8}$$
$41$ $$( 2592 - 944 T - 96 T^{2} + 12 T^{3} + T^{4} )^{2}$$
$43$ $$( -864 - 432 T + 24 T^{2} + 16 T^{3} + T^{4} )^{2}$$
$47$ $$3686400 + 653312 T^{2} + 24576 T^{4} + 288 T^{6} + T^{8}$$
$53$ $$( -2544 - 128 T + 152 T^{2} + 24 T^{3} + T^{4} )^{2}$$
$59$ $$T^{8}$$
$61$ $$( 2144 + 176 T - 112 T^{2} - 4 T^{3} + T^{4} )^{2}$$
$67$ $$10419984 + 1036640 T^{2} + 29688 T^{4} + 304 T^{6} + T^{8}$$
$71$ $$( 768 + 256 T - 64 T^{2} - 8 T^{3} + T^{4} )^{2}$$
$73$ $$( 2480 - 592 T - 128 T^{2} + 8 T^{3} + T^{4} )^{2}$$
$79$ $$18455616 + 1894112 T^{2} + 43328 T^{4} + 360 T^{6} + T^{8}$$
$83$ $$68757264 + 3420320 T^{2} + 59608 T^{4} + 424 T^{6} + T^{8}$$
$89$ $$( 9312 + 208 T - 272 T^{2} - 4 T^{3} + T^{4} )^{2}$$
$97$ $$2322576 + 429920 T^{2} + 22328 T^{4} + 288 T^{6} + T^{8}$$