# Properties

 Label 2394.2.b.j 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: $$x^{8} + 10x^{6} + 30x^{4} + 24x^{2} + 1$$ 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 + b3 * q^5 - q^7 + q^8 $$q + q^{2} + q^{4} + \beta_{3} q^{5} - q^{7} + q^{8} + \beta_{3} q^{10} - \beta_{5} q^{11} + (\beta_{7} - \beta_{6} - \beta_{5}) q^{13} - q^{14} + q^{16} - \beta_{5} q^{17} + (\beta_{7} - \beta_{6} + \beta_{4} + 1) q^{19} + \beta_{3} q^{20} - \beta_{5} q^{22} + (\beta_{5} - \beta_{2}) q^{23} + (\beta_{7} + \beta_{4} - 3) q^{25} + (\beta_{7} - \beta_{6} - \beta_{5}) q^{26} - q^{28} + (\beta_{7} + \beta_{4} - 2) q^{29} + ( - \beta_{5} - \beta_{3}) q^{31} + q^{32} - \beta_{5} q^{34} - \beta_{3} q^{35} + (2 \beta_{3} - \beta_{2}) q^{37} + (\beta_{7} - \beta_{6} + \beta_{4} + 1) q^{38} + \beta_{3} q^{40} + (2 \beta_{6} - 2 \beta_{4} - \beta_1 + 5) q^{41} + ( - \beta_{6} + \beta_{4} - \beta_1 - 5) q^{43} - \beta_{5} q^{44} + (\beta_{5} - \beta_{2}) q^{46} + ( - 2 \beta_{7} + 2 \beta_{6} + 2 \beta_{5}) q^{47} + q^{49} + (\beta_{7} + \beta_{4} - 3) q^{50} + (\beta_{7} - \beta_{6} - \beta_{5}) q^{52} + ( - \beta_{7} - \beta_{4} - 2 \beta_1 + 6) q^{53} + ( - \beta_{7} - \beta_{6} - 3 \beta_1 - 1) q^{55} - q^{56} + (\beta_{7} + \beta_{4} - 2) q^{58} + ( - 2 \beta_{7} - 2 \beta_{4} - \beta_1 + 1) q^{61} + ( - \beta_{5} - \beta_{3}) q^{62} + q^{64} + (\beta_{7} - 3 \beta_{6} + 4 \beta_{4} + \beta_1 - 1) q^{65} + (2 \beta_{7} - 2 \beta_{6} - \beta_{5} - \beta_{2}) q^{67} - \beta_{5} q^{68} - \beta_{3} q^{70} + ( - 2 \beta_1 - 2) q^{71} + (2 \beta_{7} + \beta_{6} + \beta_{4} + \beta_1 - 1) q^{73} + (2 \beta_{3} - \beta_{2}) q^{74} + (\beta_{7} - \beta_{6} + \beta_{4} + 1) q^{76} + \beta_{5} q^{77} + (\beta_{7} - \beta_{6} - \beta_{5} + 2 \beta_{3} - 2 \beta_{2}) q^{79} + \beta_{3} q^{80} + (2 \beta_{6} - 2 \beta_{4} - \beta_1 + 5) q^{82} + ( - \beta_{7} + \beta_{6} - \beta_{5} + \beta_{3} + 2 \beta_{2}) q^{83} + ( - \beta_{7} - \beta_{6} - 3 \beta_1 - 1) q^{85} + ( - \beta_{6} + \beta_{4} - \beta_1 - 5) q^{86} - \beta_{5} q^{88} + ( - 2 \beta_{6} + 2 \beta_{4} + 3 \beta_1 - 3) q^{89} + ( - \beta_{7} + \beta_{6} + \beta_{5}) q^{91} + (\beta_{5} - \beta_{2}) q^{92} + ( - 2 \beta_{7} + 2 \beta_{6} + 2 \beta_{5}) q^{94} + ( - \beta_{7} + \beta_{6} + 3 \beta_{5} + 2 \beta_{4} - \beta_{2} + 2 \beta_1) q^{95} + (2 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} + \beta_{3}) q^{97} + q^{98}+O(q^{100})$$ q + q^2 + q^4 + b3 * q^5 - q^7 + q^8 + b3 * q^10 - b5 * q^11 + (b7 - b6 - b5) * q^13 - q^14 + q^16 - b5 * q^17 + (b7 - b6 + b4 + 1) * q^19 + b3 * q^20 - b5 * q^22 + (b5 - b2) * q^23 + (b7 + b4 - 3) * q^25 + (b7 - b6 - b5) * q^26 - q^28 + (b7 + b4 - 2) * q^29 + (-b5 - b3) * q^31 + q^32 - b5 * q^34 - b3 * q^35 + (2*b3 - b2) * q^37 + (b7 - b6 + b4 + 1) * q^38 + b3 * q^40 + (2*b6 - 2*b4 - b1 + 5) * q^41 + (-b6 + b4 - b1 - 5) * q^43 - b5 * q^44 + (b5 - b2) * q^46 + (-2*b7 + 2*b6 + 2*b5) * q^47 + q^49 + (b7 + b4 - 3) * q^50 + (b7 - b6 - b5) * q^52 + (-b7 - b4 - 2*b1 + 6) * q^53 + (-b7 - b6 - 3*b1 - 1) * q^55 - q^56 + (b7 + b4 - 2) * q^58 + (-2*b7 - 2*b4 - b1 + 1) * q^61 + (-b5 - b3) * q^62 + q^64 + (b7 - 3*b6 + 4*b4 + b1 - 1) * q^65 + (2*b7 - 2*b6 - b5 - b2) * q^67 - b5 * q^68 - b3 * q^70 + (-2*b1 - 2) * q^71 + (2*b7 + b6 + b4 + b1 - 1) * q^73 + (2*b3 - b2) * q^74 + (b7 - b6 + b4 + 1) * q^76 + b5 * q^77 + (b7 - b6 - b5 + 2*b3 - 2*b2) * q^79 + b3 * q^80 + (2*b6 - 2*b4 - b1 + 5) * q^82 + (-b7 + b6 - b5 + b3 + 2*b2) * q^83 + (-b7 - b6 - 3*b1 - 1) * q^85 + (-b6 + b4 - b1 - 5) * q^86 - b5 * q^88 + (-2*b6 + 2*b4 + 3*b1 - 3) * q^89 + (-b7 + b6 + b5) * q^91 + (b5 - b2) * q^92 + (-2*b7 + 2*b6 + 2*b5) * q^94 + (-b7 + b6 + 3*b5 + 2*b4 - b2 + 2*b1) * q^95 + (2*b7 - 2*b6 - 2*b5 + b3) * q^97 + q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 8 q^{2} + 8 q^{4} - 8 q^{7} + 8 q^{8}+O(q^{10})$$ 8 * q + 8 * q^2 + 8 * q^4 - 8 * q^7 + 8 * q^8 $$8 q + 8 q^{2} + 8 q^{4} - 8 q^{7} + 8 q^{8} - 8 q^{14} + 8 q^{16} + 12 q^{19} - 24 q^{25} - 8 q^{28} - 16 q^{29} + 8 q^{32} + 12 q^{38} + 24 q^{41} - 32 q^{43} + 8 q^{49} - 24 q^{50} + 48 q^{53} - 8 q^{56} - 16 q^{58} + 8 q^{61} + 8 q^{64} + 16 q^{65} - 16 q^{71} - 16 q^{73} + 12 q^{76} + 24 q^{82} - 32 q^{86} - 8 q^{89} + 8 q^{95} + 8 q^{98}+O(q^{100})$$ 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

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

 $$\beta_{1}$$ $$=$$ $$2\nu^{4} + 10\nu^{2} + 5$$ 2*v^4 + 10*v^2 + 5 $$\beta_{2}$$ $$=$$ $$-3\nu^{5} - 18\nu^{3} - 21\nu$$ -3*v^5 - 18*v^3 - 21*v $$\beta_{3}$$ $$=$$ $$\nu^{7} + 8\nu^{5} + 19\nu^{3} + 14\nu$$ v^7 + 8*v^5 + 19*v^3 + 14*v $$\beta_{4}$$ $$=$$ $$\nu^{7} + \nu^{6} + 10\nu^{5} + 5\nu^{4} + 29\nu^{3} + 3\nu^{2} + 20\nu + 1$$ v^7 + v^6 + 10*v^5 + 5*v^4 + 29*v^3 + 3*v^2 + 20*v + 1 $$\beta_{5}$$ $$=$$ $$-\nu^{7} - 11\nu^{5} - 35\nu^{3} - 25\nu$$ -v^7 - 11*v^5 - 35*v^3 - 25*v $$\beta_{6}$$ $$=$$ $$\nu^{7} - \nu^{6} + 10\nu^{5} - 7\nu^{4} + 29\nu^{3} - 11\nu^{2} + 20\nu - 1$$ v^7 - v^6 + 10*v^5 - 7*v^4 + 29*v^3 - 11*v^2 + 20*v - 1 $$\beta_{7}$$ $$=$$ $$-\nu^{7} - \nu^{6} - 10\nu^{5} - 7\nu^{4} - 29\nu^{3} - 11\nu^{2} - 20\nu - 1$$ -v^7 - v^6 - 10*v^5 - 7*v^4 - 29*v^3 - 11*v^2 - 20*v - 1
 $$\nu$$ $$=$$ $$( -3\beta_{7} + 3\beta_{6} + 6\beta_{5} - 2\beta_{2} ) / 12$$ (-3*b7 + 3*b6 + 6*b5 - 2*b2) / 12 $$\nu^{2}$$ $$=$$ $$( \beta_{7} + \beta_{4} + \beta _1 - 5 ) / 2$$ (b7 + b4 + b1 - 5) / 2 $$\nu^{3}$$ $$=$$ $$( 15\beta_{7} - 15\beta_{6} - 24\beta_{5} + 6\beta_{3} + 4\beta_{2} ) / 12$$ (15*b7 - 15*b6 - 24*b5 + 6*b3 + 4*b2) / 12 $$\nu^{4}$$ $$=$$ $$( -5\beta_{7} - 5\beta_{4} - 4\beta _1 + 20 ) / 2$$ (-5*b7 - 5*b4 - 4*b1 + 20) / 2 $$\nu^{5}$$ $$=$$ $$( -69\beta_{7} + 69\beta_{6} + 102\beta_{5} - 36\beta_{3} - 14\beta_{2} ) / 12$$ (-69*b7 + 69*b6 + 102*b5 - 36*b3 - 14*b2) / 12 $$\nu^{6}$$ $$=$$ $$( 23\beta_{7} - \beta_{6} + 24\beta_{4} + 17\beta _1 - 87 ) / 2$$ (23*b7 - b6 + 24*b4 + 17*b1 - 87) / 2 $$\nu^{7}$$ $$=$$ $$( 309\beta_{7} - 309\beta_{6} - 444\beta_{5} + 186\beta_{3} + 64\beta_{2} ) / 12$$ (309*b7 - 309*b6 - 444*b5 + 186*b3 + 64*b2) / 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.j yes 8
3.b odd 2 1 2394.2.b.i 8
19.b odd 2 1 2394.2.b.i 8
57.d even 2 1 inner 2394.2.b.j yes 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2394.2.b.i 8 3.b odd 2 1
2394.2.b.i 8 19.b odd 2 1
2394.2.b.j yes 8 1.a even 1 1 trivial
2394.2.b.j yes 8 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 $$S_{2}^{\mathrm{new}}(2394, [\chi])$$:

 $$T_{5}^{8} + 32T_{5}^{6} + 344T_{5}^{4} + 1376T_{5}^{2} + 1296$$ T5^8 + 32*T5^6 + 344*T5^4 + 1376*T5^2 + 1296 $$T_{29}^{4} + 8T_{29}^{3} - 16T_{29}^{2} - 160T_{29} - 192$$ T29^4 + 8*T29^3 - 16*T29^2 - 160*T29 - 192 $$T_{53}^{4} - 24T_{53}^{3} + 152T_{53}^{2} + 128T_{53} - 2544$$ T53^4 - 24*T53^3 + 152*T53^2 + 128*T53 - 2544

## Hecke characteristic polynomials

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