# Properties

 Label 2790.2.d.i Level $2790$ Weight $2$ Character orbit 2790.d Analytic conductor $22.278$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$2790 = 2 \cdot 3^{2} \cdot 5 \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2790.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$22.2782621639$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 930) 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.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} - q^{4} + ( - \beta_1 - 2) q^{5} + (\beta_{2} + 2 \beta_1) q^{7} + \beta_1 q^{8}+O(q^{10})$$ q - b1 * q^2 - q^4 + (-b1 - 2) * q^5 + (b2 + 2*b1) * q^7 + b1 * q^8 $$q - \beta_1 q^{2} - q^{4} + ( - \beta_1 - 2) q^{5} + (\beta_{2} + 2 \beta_1) q^{7} + \beta_1 q^{8} + (2 \beta_1 - 1) q^{10} + ( - \beta_{3} - 2) q^{11} + ( - \beta_{2} + 2 \beta_1) q^{13} + (\beta_{3} + 2) q^{14} + q^{16} + ( - \beta_{2} - 2 \beta_1) q^{17} + (\beta_1 + 2) q^{20} + (\beta_{2} + 2 \beta_1) q^{22} + 3 \beta_{2} q^{23} + (4 \beta_1 + 3) q^{25} + ( - \beta_{3} + 2) q^{26} + ( - \beta_{2} - 2 \beta_1) q^{28} + ( - 2 \beta_{3} + 4) q^{29} + q^{31} - \beta_1 q^{32} + ( - \beta_{3} - 2) q^{34} + (\beta_{3} - 2 \beta_{2} - 4 \beta_1 + 2) q^{35} + ( - 3 \beta_{2} + 2 \beta_1) q^{37} + ( - 2 \beta_1 + 1) q^{40} + (2 \beta_{3} - 2) q^{41} + (2 \beta_{2} - 4 \beta_1) q^{43} + (\beta_{3} + 2) q^{44} + 3 \beta_{3} q^{46} + 2 \beta_{2} q^{47} + ( - 4 \beta_{3} - 5) q^{49} + ( - 3 \beta_1 + 4) q^{50} + (\beta_{2} - 2 \beta_1) q^{52} + ( - 2 \beta_{2} - 6 \beta_1) q^{53} + (2 \beta_{3} + \beta_{2} + 2 \beta_1 + 4) q^{55} + ( - \beta_{3} - 2) q^{56} + (2 \beta_{2} - 4 \beta_1) q^{58} + (\beta_{3} + 6) q^{59} + ( - \beta_{3} - 2) q^{61} - \beta_1 q^{62} - q^{64} + ( - \beta_{3} + 2 \beta_{2} - 4 \beta_1 + 2) q^{65} + ( - \beta_{2} - 12 \beta_1) q^{67} + (\beta_{2} + 2 \beta_1) q^{68} + ( - 2 \beta_{3} - \beta_{2} - 2 \beta_1 - 4) q^{70} - \beta_{3} q^{71} + (2 \beta_{2} - 8 \beta_1) q^{73} + ( - 3 \beta_{3} + 2) q^{74} + ( - 4 \beta_{2} - 12 \beta_1) q^{77} + 4 \beta_{3} q^{79} + ( - \beta_1 - 2) q^{80} + ( - 2 \beta_{2} + 2 \beta_1) q^{82} + ( - 2 \beta_{2} - 4 \beta_1) q^{83} + ( - \beta_{3} + 2 \beta_{2} + 4 \beta_1 - 2) q^{85} + (2 \beta_{3} - 4) q^{86} + ( - \beta_{2} - 2 \beta_1) q^{88} + ( - \beta_{3} + 2) q^{89} + 4 q^{91} - 3 \beta_{2} q^{92} + 2 \beta_{3} q^{94} - 4 \beta_{2} q^{97} + (4 \beta_{2} + 5 \beta_1) q^{98}+O(q^{100})$$ q - b1 * q^2 - q^4 + (-b1 - 2) * q^5 + (b2 + 2*b1) * q^7 + b1 * q^8 + (2*b1 - 1) * q^10 + (-b3 - 2) * q^11 + (-b2 + 2*b1) * q^13 + (b3 + 2) * q^14 + q^16 + (-b2 - 2*b1) * q^17 + (b1 + 2) * q^20 + (b2 + 2*b1) * q^22 + 3*b2 * q^23 + (4*b1 + 3) * q^25 + (-b3 + 2) * q^26 + (-b2 - 2*b1) * q^28 + (-2*b3 + 4) * q^29 + q^31 - b1 * q^32 + (-b3 - 2) * q^34 + (b3 - 2*b2 - 4*b1 + 2) * q^35 + (-3*b2 + 2*b1) * q^37 + (-2*b1 + 1) * q^40 + (2*b3 - 2) * q^41 + (2*b2 - 4*b1) * q^43 + (b3 + 2) * q^44 + 3*b3 * q^46 + 2*b2 * q^47 + (-4*b3 - 5) * q^49 + (-3*b1 + 4) * q^50 + (b2 - 2*b1) * q^52 + (-2*b2 - 6*b1) * q^53 + (2*b3 + b2 + 2*b1 + 4) * q^55 + (-b3 - 2) * q^56 + (2*b2 - 4*b1) * q^58 + (b3 + 6) * q^59 + (-b3 - 2) * q^61 - b1 * q^62 - q^64 + (-b3 + 2*b2 - 4*b1 + 2) * q^65 + (-b2 - 12*b1) * q^67 + (b2 + 2*b1) * q^68 + (-2*b3 - b2 - 2*b1 - 4) * q^70 - b3 * q^71 + (2*b2 - 8*b1) * q^73 + (-3*b3 + 2) * q^74 + (-4*b2 - 12*b1) * q^77 + 4*b3 * q^79 + (-b1 - 2) * q^80 + (-2*b2 + 2*b1) * q^82 + (-2*b2 - 4*b1) * q^83 + (-b3 + 2*b2 + 4*b1 - 2) * q^85 + (2*b3 - 4) * q^86 + (-b2 - 2*b1) * q^88 + (-b3 + 2) * q^89 + 4 * q^91 - 3*b2 * q^92 + 2*b3 * q^94 - 4*b2 * q^97 + (4*b2 + 5*b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4} - 8 q^{5}+O(q^{10})$$ 4 * q - 4 * q^4 - 8 * q^5 $$4 q - 4 q^{4} - 8 q^{5} - 4 q^{10} - 8 q^{11} + 8 q^{14} + 4 q^{16} + 8 q^{20} + 12 q^{25} + 8 q^{26} + 16 q^{29} + 4 q^{31} - 8 q^{34} + 8 q^{35} + 4 q^{40} - 8 q^{41} + 8 q^{44} - 20 q^{49} + 16 q^{50} + 16 q^{55} - 8 q^{56} + 24 q^{59} - 8 q^{61} - 4 q^{64} + 8 q^{65} - 16 q^{70} + 8 q^{74} - 8 q^{80} - 8 q^{85} - 16 q^{86} + 8 q^{89} + 16 q^{91}+O(q^{100})$$ 4 * q - 4 * q^4 - 8 * q^5 - 4 * q^10 - 8 * q^11 + 8 * q^14 + 4 * q^16 + 8 * q^20 + 12 * q^25 + 8 * q^26 + 16 * q^29 + 4 * q^31 - 8 * q^34 + 8 * q^35 + 4 * q^40 - 8 * q^41 + 8 * q^44 - 20 * q^49 + 16 * q^50 + 16 * q^55 - 8 * q^56 + 24 * q^59 - 8 * q^61 - 4 * q^64 + 8 * q^65 - 16 * q^70 + 8 * q^74 - 8 * q^80 - 8 * q^85 - 16 * q^86 + 8 * q^89 + 16 * q^91

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{8}^{2}$$ v^2 $$\beta_{2}$$ $$=$$ $$2\zeta_{8}^{3} + 2\zeta_{8}$$ 2*v^3 + 2*v $$\beta_{3}$$ $$=$$ $$-2\zeta_{8}^{3} + 2\zeta_{8}$$ -2*v^3 + 2*v
 $$\zeta_{8}$$ $$=$$ $$( \beta_{3} + \beta_{2} ) / 4$$ (b3 + b2) / 4 $$\zeta_{8}^{2}$$ $$=$$ $$\beta_1$$ b1 $$\zeta_{8}^{3}$$ $$=$$ $$( -\beta_{3} + \beta_{2} ) / 4$$ (-b3 + b2) / 4

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2790\mathbb{Z}\right)^\times$$.

 $$n$$ $$1117$$ $$1801$$ $$2171$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
559.1
 −0.707107 − 0.707107i 0.707107 + 0.707107i 0.707107 − 0.707107i −0.707107 + 0.707107i
1.00000i 0 −1.00000 −2.00000 1.00000i 0 0.828427i 1.00000i 0 −1.00000 + 2.00000i
559.2 1.00000i 0 −1.00000 −2.00000 1.00000i 0 4.82843i 1.00000i 0 −1.00000 + 2.00000i
559.3 1.00000i 0 −1.00000 −2.00000 + 1.00000i 0 4.82843i 1.00000i 0 −1.00000 2.00000i
559.4 1.00000i 0 −1.00000 −2.00000 + 1.00000i 0 0.828427i 1.00000i 0 −1.00000 2.00000i
 $$n$$: 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
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2790.2.d.i 4
3.b odd 2 1 930.2.d.g 4
5.b even 2 1 inner 2790.2.d.i 4
15.d odd 2 1 930.2.d.g 4
15.e even 4 1 4650.2.a.cc 2
15.e even 4 1 4650.2.a.cf 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.d.g 4 3.b odd 2 1
930.2.d.g 4 15.d odd 2 1
2790.2.d.i 4 1.a even 1 1 trivial
2790.2.d.i 4 5.b even 2 1 inner
4650.2.a.cc 2 15.e even 4 1
4650.2.a.cf 2 15.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2790, [\chi])$$:

 $$T_{7}^{4} + 24T_{7}^{2} + 16$$ T7^4 + 24*T7^2 + 16 $$T_{11}^{2} + 4T_{11} - 4$$ T11^2 + 4*T11 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 1)^{2}$$
$3$ $$T^{4}$$
$5$ $$(T^{2} + 4 T + 5)^{2}$$
$7$ $$T^{4} + 24T^{2} + 16$$
$11$ $$(T^{2} + 4 T - 4)^{2}$$
$13$ $$T^{4} + 24T^{2} + 16$$
$17$ $$T^{4} + 24T^{2} + 16$$
$19$ $$T^{4}$$
$23$ $$(T^{2} + 72)^{2}$$
$29$ $$(T^{2} - 8 T - 16)^{2}$$
$31$ $$(T - 1)^{4}$$
$37$ $$T^{4} + 152T^{2} + 4624$$
$41$ $$(T^{2} + 4 T - 28)^{2}$$
$43$ $$T^{4} + 96T^{2} + 256$$
$47$ $$(T^{2} + 32)^{2}$$
$53$ $$T^{4} + 136T^{2} + 16$$
$59$ $$(T^{2} - 12 T + 28)^{2}$$
$61$ $$(T^{2} + 4 T - 4)^{2}$$
$67$ $$T^{4} + 304 T^{2} + 18496$$
$71$ $$(T^{2} - 8)^{2}$$
$73$ $$T^{4} + 192T^{2} + 1024$$
$79$ $$(T^{2} - 128)^{2}$$
$83$ $$T^{4} + 96T^{2} + 256$$
$89$ $$(T^{2} - 4 T - 4)^{2}$$
$97$ $$(T^{2} + 128)^{2}$$
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