# Properties

 Label 1805.2.b.h Level $1805$ Weight $2$ Character orbit 1805.b Analytic conductor $14.413$ Analytic rank $0$ Dimension $8$ CM discriminant -95 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1805 = 5 \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1805.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$14.4129975648$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.280944640000.2 Defining polynomial: $$x^{8} + 16x^{6} + 80x^{4} + 128x^{2} + 19$$ x^8 + 16*x^6 + 80*x^4 + 128*x^2 + 19 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + \beta_{5} q^{3} + (\beta_{2} - 2) q^{4} + \beta_{3} q^{5} + (\beta_{6} - \beta_{3}) q^{6} + (\beta_{4} - 2 \beta_1) q^{8} + ( - \beta_{6} - \beta_{2} - 3) q^{9}+O(q^{10})$$ q + b1 * q^2 + b5 * q^3 + (b2 - 2) * q^4 + b3 * q^5 + (b6 - b3) * q^6 + (b4 - 2*b1) * q^8 + (-b6 - b2 - 3) * q^9 $$q + \beta_1 q^{2} + \beta_{5} q^{3} + (\beta_{2} - 2) q^{4} + \beta_{3} q^{5} + (\beta_{6} - \beta_{3}) q^{6} + (\beta_{4} - 2 \beta_1) q^{8} + ( - \beta_{6} - \beta_{2} - 3) q^{9} + (\beta_{5} - \beta_{4}) q^{10} + (\beta_{6} - \beta_{2}) q^{11} + (\beta_{7} - \beta_{5} + \beta_{4}) q^{12} + (\beta_{7} - \beta_1) q^{13} + (\beta_{7} + \beta_1) q^{15} + (3 \beta_{3} - 2 \beta_{2} + 4) q^{16} + ( - \beta_{7} + 2 \beta_{5} - \beta_{4} - \beta_1) q^{18} + (\beta_{6} - 2 \beta_{3} + 2 \beta_{2}) q^{20} + (\beta_{7} - 2 \beta_{5} - \beta_{4} + 2 \beta_1) q^{22} + ( - \beta_{6} + 2 \beta_{3} - 2 \beta_{2} - 1) q^{24} + 5 q^{25} + ( - 2 \beta_{6} - \beta_{2} + 3) q^{26} + ( - \beta_{7} - 3 \beta_{5} + 3 \beta_1) q^{27} + ( - 2 \beta_{6} + \beta_{2} - 5) q^{30} + (3 \beta_{5} - 3 \beta_{4} + 4 \beta_1) q^{32} + ( - \beta_{7} + \beta_{5} - 2 \beta_{4} - 3 \beta_1) q^{33} + (2 \beta_{6} - 5 \beta_{3} - \beta_{2} - 1) q^{36} + ( - \beta_{5} - 2 \beta_{4}) q^{37} + ( - \beta_{6} - 4 \beta_{3} - 3 \beta_{2}) q^{39} + (\beta_{7} - 2 \beta_{5} + 2 \beta_{4} - 4 \beta_1) q^{40} + ( - 2 \beta_{6} - \beta_{3} + 2 \beta_{2} - 9) q^{44} + (\beta_{6} - 3 \beta_{3} - 3 \beta_{2}) q^{45} + (\beta_{7} + 2 \beta_{5} - 2 \beta_{4} + 3 \beta_1) q^{48} + 7 q^{49} + 5 \beta_1 q^{50} + (4 \beta_{5} - \beta_{4} + 3 \beta_1) q^{52} + (3 \beta_{5} - 2 \beta_{4}) q^{53} + ( - \beta_{6} + 3 \beta_{3} + 3 \beta_{2} - 11) q^{54} + ( - 3 \beta_{6} - \beta_{2}) q^{55} + (4 \beta_{5} + \beta_{4} - 5 \beta_1) q^{60} + (3 \beta_{6} - \beta_{2}) q^{61} + (3 \beta_{6} - 6 \beta_{3} + 6 \beta_{2} - 8) q^{64} + (3 \beta_{5} + 2 \beta_{4}) q^{65} + (3 \beta_{6} - 7 \beta_{3} + \beta_{2} + 13) q^{66} + ( - \beta_{7} - 5 \beta_1) q^{67} + ( - 5 \beta_{5} + 2 \beta_{4} - \beta_1) q^{72} + ( - \beta_{6} - 5 \beta_{3} + 4 \beta_{2}) q^{74} + 5 \beta_{5} q^{75} + ( - \beta_{7} - 2 \beta_{5} + \beta_{4} + 6 \beta_1) q^{78} + ( - 2 \beta_{6} + 4 \beta_{3} - 4 \beta_{2} + 15) q^{80} + (3 \beta_{6} + 2 \beta_{3} + 3 \beta_{2} + 9) q^{81} + ( - \beta_{5} + \beta_{4} - 9 \beta_1) q^{88} + (\beta_{7} - 5 \beta_{5} + 6 \beta_1) q^{90} + ( - 2 \beta_{6} - 4 \beta_{3} + 3 \beta_{2} - 15) q^{96} + (5 \beta_{5} - 2 \beta_{4}) q^{97} + 7 \beta_1 q^{98} + ( - 3 \beta_{6} + 8 \beta_{3} + 3 \beta_{2} - 4) q^{99}+O(q^{100})$$ q + b1 * q^2 + b5 * q^3 + (b2 - 2) * q^4 + b3 * q^5 + (b6 - b3) * q^6 + (b4 - 2*b1) * q^8 + (-b6 - b2 - 3) * q^9 + (b5 - b4) * q^10 + (b6 - b2) * q^11 + (b7 - b5 + b4) * q^12 + (b7 - b1) * q^13 + (b7 + b1) * q^15 + (3*b3 - 2*b2 + 4) * q^16 + (-b7 + 2*b5 - b4 - b1) * q^18 + (b6 - 2*b3 + 2*b2) * q^20 + (b7 - 2*b5 - b4 + 2*b1) * q^22 + (-b6 + 2*b3 - 2*b2 - 1) * q^24 + 5 * q^25 + (-2*b6 - b2 + 3) * q^26 + (-b7 - 3*b5 + 3*b1) * q^27 + (-2*b6 + b2 - 5) * q^30 + (3*b5 - 3*b4 + 4*b1) * q^32 + (-b7 + b5 - 2*b4 - 3*b1) * q^33 + (2*b6 - 5*b3 - b2 - 1) * q^36 + (-b5 - 2*b4) * q^37 + (-b6 - 4*b3 - 3*b2) * q^39 + (b7 - 2*b5 + 2*b4 - 4*b1) * q^40 + (-2*b6 - b3 + 2*b2 - 9) * q^44 + (b6 - 3*b3 - 3*b2) * q^45 + (b7 + 2*b5 - 2*b4 + 3*b1) * q^48 + 7 * q^49 + 5*b1 * q^50 + (4*b5 - b4 + 3*b1) * q^52 + (3*b5 - 2*b4) * q^53 + (-b6 + 3*b3 + 3*b2 - 11) * q^54 + (-3*b6 - b2) * q^55 + (4*b5 + b4 - 5*b1) * q^60 + (3*b6 - b2) * q^61 + (3*b6 - 6*b3 + 6*b2 - 8) * q^64 + (3*b5 + 2*b4) * q^65 + (3*b6 - 7*b3 + b2 + 13) * q^66 + (-b7 - 5*b1) * q^67 + (-5*b5 + 2*b4 - b1) * q^72 + (-b6 - 5*b3 + 4*b2) * q^74 + 5*b5 * q^75 + (-b7 - 2*b5 + b4 + 6*b1) * q^78 + (-2*b6 + 4*b3 - 4*b2 + 15) * q^80 + (3*b6 + 2*b3 + 3*b2 + 9) * q^81 + (-b5 + b4 - 9*b1) * q^88 + (b7 - 5*b5 + 6*b1) * q^90 + (-2*b6 - 4*b3 + 3*b2 - 15) * q^96 + (5*b5 - 2*b4) * q^97 + 7*b1 * q^98 + (-3*b6 + 8*b3 + 3*b2 - 4) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 16 q^{4} - 24 q^{9}+O(q^{10})$$ 8 * q - 16 * q^4 - 24 * q^9 $$8 q - 16 q^{4} - 24 q^{9} + 32 q^{16} - 8 q^{24} + 40 q^{25} + 24 q^{26} - 40 q^{30} - 8 q^{36} - 72 q^{44} + 56 q^{49} - 88 q^{54} - 64 q^{64} + 104 q^{66} + 120 q^{80} + 72 q^{81} - 120 q^{96} - 32 q^{99}+O(q^{100})$$ 8 * q - 16 * q^4 - 24 * q^9 + 32 * q^16 - 8 * q^24 + 40 * q^25 + 24 * q^26 - 40 * q^30 - 8 * q^36 - 72 * q^44 + 56 * q^49 - 88 * q^54 - 64 * q^64 + 104 * q^66 + 120 * q^80 + 72 * q^81 - 120 * q^96 - 32 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 16x^{6} + 80x^{4} + 128x^{2} + 19$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} + 4$$ v^2 + 4 $$\beta_{3}$$ $$=$$ $$( \nu^{4} + 8\nu^{2} + 8 ) / 3$$ (v^4 + 8*v^2 + 8) / 3 $$\beta_{4}$$ $$=$$ $$\nu^{3} + 6\nu$$ v^3 + 6*v $$\beta_{5}$$ $$=$$ $$( \nu^{5} + 11\nu^{3} + 26\nu ) / 3$$ (v^5 + 11*v^3 + 26*v) / 3 $$\beta_{6}$$ $$=$$ $$( \nu^{6} + 12\nu^{4} + 34\nu^{2} + 8 ) / 3$$ (v^6 + 12*v^4 + 34*v^2 + 8) / 3 $$\beta_{7}$$ $$=$$ $$( \nu^{7} + 14\nu^{5} + 56\nu^{3} + 60\nu ) / 3$$ (v^7 + 14*v^5 + 56*v^3 + 60*v) / 3
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} - 4$$ b2 - 4 $$\nu^{3}$$ $$=$$ $$\beta_{4} - 6\beta_1$$ b4 - 6*b1 $$\nu^{4}$$ $$=$$ $$3\beta_{3} - 8\beta_{2} + 24$$ 3*b3 - 8*b2 + 24 $$\nu^{5}$$ $$=$$ $$3\beta_{5} - 11\beta_{4} + 40\beta_1$$ 3*b5 - 11*b4 + 40*b1 $$\nu^{6}$$ $$=$$ $$3\beta_{6} - 36\beta_{3} + 62\beta_{2} - 160$$ 3*b6 - 36*b3 + 62*b2 - 160 $$\nu^{7}$$ $$=$$ $$3\beta_{7} - 42\beta_{5} + 98\beta_{4} - 284\beta_1$$ 3*b7 - 42*b5 + 98*b4 - 284*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1805\mathbb{Z}\right)^\times$$.

 $$n$$ $$362$$ $$1446$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1084.1
 − 2.79913i − 2.26640i − 1.69217i − 0.406045i 0.406045i 1.69217i 2.26640i 2.79913i
2.79913i 1.12228i −5.83513 2.23607 −3.14142 0 10.7350i 1.74048 6.25904i
1084.2 2.26640i 3.11095i −3.13657 −2.23607 7.05066 0 2.57593i −6.67802 5.06783i
1084.3 1.69217i 1.52380i −0.863428 −2.23607 −2.57853 0 1.92327i 0.678024 3.78380i
1084.4 0.406045i 3.27727i 1.83513 2.23607 −1.33072 0 1.55723i −7.74048 0.907944i
1084.5 0.406045i 3.27727i 1.83513 2.23607 −1.33072 0 1.55723i −7.74048 0.907944i
1084.6 1.69217i 1.52380i −0.863428 −2.23607 −2.57853 0 1.92327i 0.678024 3.78380i
1084.7 2.26640i 3.11095i −3.13657 −2.23607 7.05066 0 2.57593i −6.67802 5.06783i
1084.8 2.79913i 1.12228i −5.83513 2.23607 −3.14142 0 10.7350i 1.74048 6.25904i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1084.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
95.d odd 2 1 CM by $$\Q(\sqrt{-95})$$
5.b even 2 1 inner
19.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1805.2.b.h 8
5.b even 2 1 inner 1805.2.b.h 8
5.c odd 4 2 9025.2.a.cb 8
19.b odd 2 1 inner 1805.2.b.h 8
95.d odd 2 1 CM 1805.2.b.h 8
95.g even 4 2 9025.2.a.cb 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1805.2.b.h 8 1.a even 1 1 trivial
1805.2.b.h 8 5.b even 2 1 inner
1805.2.b.h 8 19.b odd 2 1 inner
1805.2.b.h 8 95.d odd 2 1 CM
9025.2.a.cb 8 5.c odd 4 2
9025.2.a.cb 8 95.g even 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1805, [\chi])$$:

 $$T_{2}^{8} + 16T_{2}^{6} + 80T_{2}^{4} + 128T_{2}^{2} + 19$$ T2^8 + 16*T2^6 + 80*T2^4 + 128*T2^2 + 19 $$T_{29}$$ T29

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} + 16 T^{6} + 80 T^{4} + 128 T^{2} + \cdots + 19$$
$3$ $$T^{8} + 24 T^{6} + 180 T^{4} + \cdots + 304$$
$5$ $$(T^{2} - 5)^{4}$$
$7$ $$T^{8}$$
$11$ $$(T^{4} - 44 T^{2} + 304)^{2}$$
$13$ $$T^{8} + 104 T^{6} + 3380 T^{4} + \cdots + 109744$$
$17$ $$T^{8}$$
$19$ $$T^{8}$$
$23$ $$T^{8}$$
$29$ $$T^{8}$$
$31$ $$T^{8}$$
$37$ $$T^{8} + 296 T^{6} + 27380 T^{4} + \cdots + 511024$$
$41$ $$T^{8}$$
$43$ $$T^{8}$$
$47$ $$T^{8}$$
$53$ $$T^{8} + 424 T^{6} + \cdots + 30935344$$
$59$ $$T^{8}$$
$61$ $$(T^{4} - 244 T^{2} + 304)^{2}$$
$67$ $$T^{8} + 536 T^{6} + \cdots + 43666864$$
$71$ $$T^{8}$$
$73$ $$T^{8}$$
$79$ $$T^{8}$$
$83$ $$T^{8}$$
$89$ $$T^{8}$$
$97$ $$T^{8} + 776 T^{6} + \cdots + 116480944$$