# Properties

 Label 912.2.f.f Level $912$ Weight $2$ Character orbit 912.f Analytic conductor $7.282$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$912 = 2^{4} \cdot 3 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 912.f (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.28235666434$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-5})$$ Defining polynomial: $$x^{4} + 4x^{2} + 9$$ x^4 + 4*x^2 + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 57) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{2} - \beta_1) q^{3} - \beta_{3} q^{5} - q^{7} + ( - \beta_{3} - 2) q^{9}+O(q^{10})$$ q + (-b2 - b1) * q^3 - b3 * q^5 - q^7 + (-b3 - 2) * q^9 $$q + ( - \beta_{2} - \beta_1) q^{3} - \beta_{3} q^{5} - q^{7} + ( - \beta_{3} - 2) q^{9} - \beta_{3} q^{11} + ( - \beta_{2} - 2 \beta_1) q^{13} + (2 \beta_{2} - \beta_1) q^{15} + 3 \beta_{3} q^{17} + ( - \beta_{2} - 2 \beta_1 - 3) q^{19} + (\beta_{2} + \beta_1) q^{21} + 2 \beta_{3} q^{23} + (4 \beta_{2} + \beta_1) q^{27} + 4 \beta_{2} q^{29} + ( - \beta_{2} - 2 \beta_1) q^{31} + (2 \beta_{2} - \beta_1) q^{33} + \beta_{3} q^{35} + ( - 3 \beta_{2} - 6 \beta_1) q^{37} + ( - \beta_{3} - 5) q^{39} - 7 \beta_{2} q^{41} + 5 q^{43} + (2 \beta_{3} - 5) q^{45} + \beta_{3} q^{47} - 6 q^{49} + ( - 6 \beta_{2} + 3 \beta_1) q^{51} + 3 \beta_{2} q^{53} - 5 q^{55} + ( - \beta_{3} + 3 \beta_{2} + 3 \beta_1 - 5) q^{57} - \beta_{2} q^{59} - q^{61} + (\beta_{3} + 2) q^{63} + 5 \beta_{2} q^{65} + (2 \beta_{2} + 4 \beta_1) q^{67} + ( - 4 \beta_{2} + 2 \beta_1) q^{69} + 3 \beta_{2} q^{71} - 3 q^{73} + \beta_{3} q^{77} + ( - 4 \beta_{2} - 8 \beta_1) q^{79} + (4 \beta_{3} - 1) q^{81} - 4 \beta_{3} q^{83} + 15 q^{85} + (4 \beta_{3} - 4) q^{87} - 9 \beta_{2} q^{89} + (\beta_{2} + 2 \beta_1) q^{91} + ( - \beta_{3} - 5) q^{93} + (3 \beta_{3} + 5 \beta_{2}) q^{95} + ( - \beta_{2} - 2 \beta_1) q^{97} + (2 \beta_{3} - 5) q^{99}+O(q^{100})$$ q + (-b2 - b1) * q^3 - b3 * q^5 - q^7 + (-b3 - 2) * q^9 - b3 * q^11 + (-b2 - 2*b1) * q^13 + (2*b2 - b1) * q^15 + 3*b3 * q^17 + (-b2 - 2*b1 - 3) * q^19 + (b2 + b1) * q^21 + 2*b3 * q^23 + (4*b2 + b1) * q^27 + 4*b2 * q^29 + (-b2 - 2*b1) * q^31 + (2*b2 - b1) * q^33 + b3 * q^35 + (-3*b2 - 6*b1) * q^37 + (-b3 - 5) * q^39 - 7*b2 * q^41 + 5 * q^43 + (2*b3 - 5) * q^45 + b3 * q^47 - 6 * q^49 + (-6*b2 + 3*b1) * q^51 + 3*b2 * q^53 - 5 * q^55 + (-b3 + 3*b2 + 3*b1 - 5) * q^57 - b2 * q^59 - q^61 + (b3 + 2) * q^63 + 5*b2 * q^65 + (2*b2 + 4*b1) * q^67 + (-4*b2 + 2*b1) * q^69 + 3*b2 * q^71 - 3 * q^73 + b3 * q^77 + (-4*b2 - 8*b1) * q^79 + (4*b3 - 1) * q^81 - 4*b3 * q^83 + 15 * q^85 + (4*b3 - 4) * q^87 - 9*b2 * q^89 + (b2 + 2*b1) * q^91 + (-b3 - 5) * q^93 + (3*b3 + 5*b2) * q^95 + (-b2 - 2*b1) * q^97 + (2*b3 - 5) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{7} - 8 q^{9}+O(q^{10})$$ 4 * q - 4 * q^7 - 8 * q^9 $$4 q - 4 q^{7} - 8 q^{9} - 12 q^{19} - 20 q^{39} + 20 q^{43} - 20 q^{45} - 24 q^{49} - 20 q^{55} - 20 q^{57} - 4 q^{61} + 8 q^{63} - 12 q^{73} - 4 q^{81} + 60 q^{85} - 16 q^{87} - 20 q^{93} - 20 q^{99}+O(q^{100})$$ 4 * q - 4 * q^7 - 8 * q^9 - 12 * q^19 - 20 * q^39 + 20 * q^43 - 20 * q^45 - 24 * q^49 - 20 * q^55 - 20 * q^57 - 4 * q^61 + 8 * q^63 - 12 * q^73 - 4 * q^81 + 60 * q^85 - 16 * q^87 - 20 * q^93 - 20 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 4x^{2} + 9$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} + \nu ) / 3$$ (v^3 + v) / 3 $$\beta_{3}$$ $$=$$ $$\nu^{2} + 2$$ v^2 + 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} - 2$$ b3 - 2 $$\nu^{3}$$ $$=$$ $$3\beta_{2} - \beta_1$$ 3*b2 - b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/912\mathbb{Z}\right)^\times$$.

 $$n$$ $$97$$ $$229$$ $$305$$ $$799$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
113.1
 −0.707107 + 1.58114i −0.707107 − 1.58114i 0.707107 + 1.58114i 0.707107 − 1.58114i
0 −0.707107 1.58114i 0 2.23607i 0 −1.00000 0 −2.00000 + 2.23607i 0
113.2 0 −0.707107 + 1.58114i 0 2.23607i 0 −1.00000 0 −2.00000 2.23607i 0
113.3 0 0.707107 1.58114i 0 2.23607i 0 −1.00000 0 −2.00000 2.23607i 0
113.4 0 0.707107 + 1.58114i 0 2.23607i 0 −1.00000 0 −2.00000 + 2.23607i 0
 $$n$$: 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
3.b odd 2 1 inner
19.b odd 2 1 inner
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 912.2.f.f 4
3.b odd 2 1 inner 912.2.f.f 4
4.b odd 2 1 57.2.d.a 4
12.b even 2 1 57.2.d.a 4
19.b odd 2 1 inner 912.2.f.f 4
57.d even 2 1 inner 912.2.f.f 4
76.d even 2 1 57.2.d.a 4
228.b odd 2 1 57.2.d.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.2.d.a 4 4.b odd 2 1
57.2.d.a 4 12.b even 2 1
57.2.d.a 4 76.d even 2 1
57.2.d.a 4 228.b odd 2 1
912.2.f.f 4 1.a even 1 1 trivial
912.2.f.f 4 3.b odd 2 1 inner
912.2.f.f 4 19.b odd 2 1 inner
912.2.f.f 4 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 $$S_{2}^{\mathrm{new}}(912, [\chi])$$:

 $$T_{5}^{2} + 5$$ T5^2 + 5 $$T_{7} + 1$$ T7 + 1 $$T_{29}^{2} - 32$$ T29^2 - 32

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + 4T^{2} + 9$$
$5$ $$(T^{2} + 5)^{2}$$
$7$ $$(T + 1)^{4}$$
$11$ $$(T^{2} + 5)^{2}$$
$13$ $$(T^{2} + 10)^{2}$$
$17$ $$(T^{2} + 45)^{2}$$
$19$ $$(T^{2} + 6 T + 19)^{2}$$
$23$ $$(T^{2} + 20)^{2}$$
$29$ $$(T^{2} - 32)^{2}$$
$31$ $$(T^{2} + 10)^{2}$$
$37$ $$(T^{2} + 90)^{2}$$
$41$ $$(T^{2} - 98)^{2}$$
$43$ $$(T - 5)^{4}$$
$47$ $$(T^{2} + 5)^{2}$$
$53$ $$(T^{2} - 18)^{2}$$
$59$ $$(T^{2} - 2)^{2}$$
$61$ $$(T + 1)^{4}$$
$67$ $$(T^{2} + 40)^{2}$$
$71$ $$(T^{2} - 18)^{2}$$
$73$ $$(T + 3)^{4}$$
$79$ $$(T^{2} + 160)^{2}$$
$83$ $$(T^{2} + 80)^{2}$$
$89$ $$(T^{2} - 162)^{2}$$
$97$ $$(T^{2} + 10)^{2}$$