Properties

 Label 8550.2.a.cv Level $8550$ Weight $2$ Character orbit 8550.a Self dual yes Analytic conductor $68.272$ Analytic rank $1$ Dimension $4$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8550.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$68.2720937282$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{24})^+$$ Defining polynomial: $$x^{4} - 4x^{2} + 1$$ x^4 - 4*x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 1710) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} + \beta_1 q^{7} + q^{8}+O(q^{10})$$ q + q^2 + q^4 + b1 * q^7 + q^8 $$q + q^{2} + q^{4} + \beta_1 q^{7} + q^{8} + \beta_{3} q^{11} - \beta_1 q^{13} + \beta_1 q^{14} + q^{16} + (\beta_{2} - 4) q^{17} - q^{19} + \beta_{3} q^{22} + ( - \beta_{2} - 2) q^{23} - \beta_1 q^{26} + \beta_1 q^{28} + ( - 3 \beta_{3} - 2 \beta_1) q^{29} + ( - 2 \beta_{2} - 4) q^{31} + q^{32} + (\beta_{2} - 4) q^{34} + ( - 2 \beta_{3} - \beta_1) q^{37} - q^{38} + ( - 2 \beta_{3} - \beta_1) q^{41} + ( - \beta_{3} - 2 \beta_1) q^{43} + \beta_{3} q^{44} + ( - \beta_{2} - 2) q^{46} + (\beta_{2} - 6) q^{47} + (2 \beta_{2} + 1) q^{49} - \beta_1 q^{52} - 6 q^{53} + \beta_1 q^{56} + ( - 3 \beta_{3} - 2 \beta_1) q^{58} + (3 \beta_{3} - \beta_1) q^{59} - 2 q^{61} + ( - 2 \beta_{2} - 4) q^{62} + q^{64} + (2 \beta_{3} + 4 \beta_1) q^{67} + (\beta_{2} - 4) q^{68} + 4 \beta_{3} q^{71} + ( - 4 \beta_{3} - 2 \beta_1) q^{73} + ( - 2 \beta_{3} - \beta_1) q^{74} - q^{76} - 4 q^{77} - 4 q^{79} + ( - 2 \beta_{3} - \beta_1) q^{82} + ( - 3 \beta_{2} - 6) q^{83} + ( - \beta_{3} - 2 \beta_1) q^{86} + \beta_{3} q^{88} + \beta_1 q^{89} + ( - 2 \beta_{2} - 8) q^{91} + ( - \beta_{2} - 2) q^{92} + (\beta_{2} - 6) q^{94} + (5 \beta_{3} + 2 \beta_1) q^{97} + (2 \beta_{2} + 1) q^{98}+O(q^{100})$$ q + q^2 + q^4 + b1 * q^7 + q^8 + b3 * q^11 - b1 * q^13 + b1 * q^14 + q^16 + (b2 - 4) * q^17 - q^19 + b3 * q^22 + (-b2 - 2) * q^23 - b1 * q^26 + b1 * q^28 + (-3*b3 - 2*b1) * q^29 + (-2*b2 - 4) * q^31 + q^32 + (b2 - 4) * q^34 + (-2*b3 - b1) * q^37 - q^38 + (-2*b3 - b1) * q^41 + (-b3 - 2*b1) * q^43 + b3 * q^44 + (-b2 - 2) * q^46 + (b2 - 6) * q^47 + (2*b2 + 1) * q^49 - b1 * q^52 - 6 * q^53 + b1 * q^56 + (-3*b3 - 2*b1) * q^58 + (3*b3 - b1) * q^59 - 2 * q^61 + (-2*b2 - 4) * q^62 + q^64 + (2*b3 + 4*b1) * q^67 + (b2 - 4) * q^68 + 4*b3 * q^71 + (-4*b3 - 2*b1) * q^73 + (-2*b3 - b1) * q^74 - q^76 - 4 * q^77 - 4 * q^79 + (-2*b3 - b1) * q^82 + (-3*b2 - 6) * q^83 + (-b3 - 2*b1) * q^86 + b3 * q^88 + b1 * q^89 + (-2*b2 - 8) * q^91 + (-b2 - 2) * q^92 + (b2 - 6) * q^94 + (5*b3 + 2*b1) * q^97 + (2*b2 + 1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{2} + 4 q^{4} + 4 q^{8}+O(q^{10})$$ 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^8 $$4 q + 4 q^{2} + 4 q^{4} + 4 q^{8} + 4 q^{16} - 16 q^{17} - 4 q^{19} - 8 q^{23} - 16 q^{31} + 4 q^{32} - 16 q^{34} - 4 q^{38} - 8 q^{46} - 24 q^{47} + 4 q^{49} - 24 q^{53} - 8 q^{61} - 16 q^{62} + 4 q^{64} - 16 q^{68} - 4 q^{76} - 16 q^{77} - 16 q^{79} - 24 q^{83} - 32 q^{91} - 8 q^{92} - 24 q^{94} + 4 q^{98}+O(q^{100})$$ 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^8 + 4 * q^16 - 16 * q^17 - 4 * q^19 - 8 * q^23 - 16 * q^31 + 4 * q^32 - 16 * q^34 - 4 * q^38 - 8 * q^46 - 24 * q^47 + 4 * q^49 - 24 * q^53 - 8 * q^61 - 16 * q^62 + 4 * q^64 - 16 * q^68 - 4 * q^76 - 16 * q^77 - 16 * q^79 - 24 * q^83 - 32 * q^91 - 8 * q^92 - 24 * q^94 + 4 * q^98

Basis of coefficient ring in terms of $$\nu = \zeta_{24} + \zeta_{24}^{-1}$$:

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

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −1.93185 −0.517638 0.517638 1.93185
1.00000 0 1.00000 0 0 −3.86370 1.00000 0 0
1.2 1.00000 0 1.00000 0 0 −1.03528 1.00000 0 0
1.3 1.00000 0 1.00000 0 0 1.03528 1.00000 0 0
1.4 1.00000 0 1.00000 0 0 3.86370 1.00000 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$1$$
$$5$$ $$-1$$
$$19$$ $$1$$

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8550.2.a.cv 4
3.b odd 2 1 8550.2.a.cu 4
5.b even 2 1 8550.2.a.cu 4
5.c odd 4 2 1710.2.d.g 8
15.d odd 2 1 inner 8550.2.a.cv 4
15.e even 4 2 1710.2.d.g 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1710.2.d.g 8 5.c odd 4 2
1710.2.d.g 8 15.e even 4 2
8550.2.a.cu 4 3.b odd 2 1
8550.2.a.cu 4 5.b even 2 1
8550.2.a.cv 4 1.a even 1 1 trivial
8550.2.a.cv 4 15.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(8550))$$:

 $$T_{7}^{4} - 16T_{7}^{2} + 16$$ T7^4 - 16*T7^2 + 16 $$T_{11}^{4} - 16T_{11}^{2} + 16$$ T11^4 - 16*T11^2 + 16 $$T_{13}^{4} - 16T_{13}^{2} + 16$$ T13^4 - 16*T13^2 + 16 $$T_{17}^{2} + 8T_{17} + 4$$ T17^2 + 8*T17 + 4 $$T_{23}^{2} + 4T_{23} - 8$$ T23^2 + 4*T23 - 8 $$T_{53} + 6$$ T53 + 6

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4} - 16T^{2} + 16$$
$11$ $$T^{4} - 16T^{2} + 16$$
$13$ $$T^{4} - 16T^{2} + 16$$
$17$ $$(T^{2} + 8 T + 4)^{2}$$
$19$ $$(T + 1)^{4}$$
$23$ $$(T^{2} + 4 T - 8)^{2}$$
$29$ $$T^{4} - 112T^{2} + 1936$$
$31$ $$(T^{2} + 8 T - 32)^{2}$$
$37$ $$T^{4} - 48T^{2} + 144$$
$41$ $$T^{4} - 48T^{2} + 144$$
$43$ $$T^{4} - 48T^{2} + 144$$
$47$ $$(T^{2} + 12 T + 24)^{2}$$
$53$ $$(T + 6)^{4}$$
$59$ $$T^{4} - 208T^{2} + 7744$$
$61$ $$(T + 2)^{4}$$
$67$ $$T^{4} - 192T^{2} + 2304$$
$71$ $$T^{4} - 256T^{2} + 4096$$
$73$ $$T^{4} - 192T^{2} + 2304$$
$79$ $$(T + 4)^{4}$$
$83$ $$(T^{2} + 12 T - 72)^{2}$$
$89$ $$T^{4} - 16T^{2} + 16$$
$97$ $$T^{4} - 304T^{2} + 1936$$