# Properties

 Label 10000.2.a.w Level $10000$ Weight $2$ Character orbit 10000.a Self dual yes Analytic conductor $79.850$ Analytic rank $1$ Dimension $4$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$10000 = 2^{4} \cdot 5^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 10000.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$79.8504020213$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{15})^+$$ Defining polynomial: $$x^{4} - x^{3} - 4x^{2} + 4x + 1$$ x^4 - x^3 - 4*x^2 + 4*x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1250) 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{3} - \beta_{2}) q^{3} + ( - 2 \beta_{3} - 2 \beta_{2}) q^{7} + (\beta_1 - 1) q^{9}+O(q^{10})$$ q + (-b3 - b2) * q^3 + (-2*b3 - 2*b2) * q^7 + (b1 - 1) * q^9 $$q + ( - \beta_{3} - \beta_{2}) q^{3} + ( - 2 \beta_{3} - 2 \beta_{2}) q^{7} + (\beta_1 - 1) q^{9} + ( - 2 \beta_{2} + 1) q^{11} + (4 \beta_{3} + 2 \beta_{2}) q^{13} + ( - 3 \beta_{3} - \beta_1 - 3) q^{17} + ( - 4 \beta_{3} - \beta_{2} + 2 \beta_1 - 6) q^{19} + (2 \beta_1 + 4) q^{21} + (4 \beta_{3} + 2 \beta_{2} - 4 \beta_1 + 4) q^{23} + (2 \beta_{3} + 3 \beta_{2} - 1) q^{27} + (6 \beta_{3} + 2 \beta_{2} - 4 \beta_1 + 6) q^{29} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 2) q^{31} + (\beta_{3} - \beta_{2} + 4) q^{33} + ( - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 4) q^{37} + (2 \beta_{3} - 4 \beta_1 - 4) q^{39} + ( - \beta_{3} + 4 \beta_{2} + \beta_1 - 1) q^{41} + ( - 5 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 6) q^{43} + (2 \beta_{3} + 4 \beta_{2} + 2 \beta_1) q^{47} + (4 \beta_1 + 1) q^{49} + (2 \beta_{3} + 4 \beta_{2} + 3 \beta_1 + 1) q^{51} + (4 \beta_{3} - 4 \beta_1 + 4) q^{53} + ( - \beta_{3} + 4 \beta_{2} + 4 \beta_1) q^{57} + ( - 3 \beta_{3} + 2 \beta_{2} + 3) q^{59} + ( - 6 \beta_{3} + 4 \beta_{2} + 2 \beta_1) q^{61} + ( - 2 \beta_{3} - 2) q^{63} + (3 \beta_{2} - 2 \beta_1 - 6) q^{67} + (6 \beta_{3} - 4 \beta_1) q^{69} + (6 \beta_{3} - 4 \beta_1 + 2) q^{71} + ( - 4 \beta_{2} + 3 \beta_1) q^{73} + (2 \beta_{3} - 2 \beta_{2} + 8) q^{77} + (4 \beta_{3} + 4 \beta_{2} + 2 \beta_1 - 2) q^{79} + (\beta_{2} - 5 \beta_1 - 3) q^{81} + (4 \beta_{3} + 2 \beta_{2} + 4 \beta_1 - 3) q^{83} + (6 \beta_{3} - 2 \beta_{2} - 6 \beta_1) q^{87} + (8 \beta_{3} + 5 \beta_{2} + 4) q^{89} + (4 \beta_{3} - 8 \beta_1 - 8) q^{91} + ( - 2 \beta_{3} - 2 \beta_1 - 6) q^{93} + ( - 9 \beta_{3} - 5) q^{97} + ( - 2 \beta_{3} + 2 \beta_{2} - \beta_1 - 1) q^{99}+O(q^{100})$$ q + (-b3 - b2) * q^3 + (-2*b3 - 2*b2) * q^7 + (b1 - 1) * q^9 + (-2*b2 + 1) * q^11 + (4*b3 + 2*b2) * q^13 + (-3*b3 - b1 - 3) * q^17 + (-4*b3 - b2 + 2*b1 - 6) * q^19 + (2*b1 + 4) * q^21 + (4*b3 + 2*b2 - 4*b1 + 4) * q^23 + (2*b3 + 3*b2 - 1) * q^27 + (6*b3 + 2*b2 - 4*b1 + 6) * q^29 + (2*b3 + 2*b2 + 2*b1 - 2) * q^31 + (b3 - b2 + 4) * q^33 + (-2*b3 + 2*b2 - 2*b1 - 4) * q^37 + (2*b3 - 4*b1 - 4) * q^39 + (-b3 + 4*b2 + b1 - 1) * q^41 + (-5*b3 - 2*b2 + 2*b1 - 6) * q^43 + (2*b3 + 4*b2 + 2*b1) * q^47 + (4*b1 + 1) * q^49 + (2*b3 + 4*b2 + 3*b1 + 1) * q^51 + (4*b3 - 4*b1 + 4) * q^53 + (-b3 + 4*b2 + 4*b1) * q^57 + (-3*b3 + 2*b2 + 3) * q^59 + (-6*b3 + 4*b2 + 2*b1) * q^61 + (-2*b3 - 2) * q^63 + (3*b2 - 2*b1 - 6) * q^67 + (6*b3 - 4*b1) * q^69 + (6*b3 - 4*b1 + 2) * q^71 + (-4*b2 + 3*b1) * q^73 + (2*b3 - 2*b2 + 8) * q^77 + (4*b3 + 4*b2 + 2*b1 - 2) * q^79 + (b2 - 5*b1 - 3) * q^81 + (4*b3 + 2*b2 + 4*b1 - 3) * q^83 + (6*b3 - 2*b2 - 6*b1) * q^87 + (8*b3 + 5*b2 + 4) * q^89 + (4*b3 - 8*b1 - 8) * q^91 + (-2*b3 - 2*b1 - 6) * q^93 + (-9*b3 - 5) * q^97 + (-2*b3 + 2*b2 - b1 - 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + q^{3} + 2 q^{7} - 3 q^{9}+O(q^{10})$$ 4 * q + q^3 + 2 * q^7 - 3 * q^9 $$4 q + q^{3} + 2 q^{7} - 3 q^{9} + 2 q^{11} - 6 q^{13} - 7 q^{17} - 15 q^{19} + 18 q^{21} + 6 q^{23} - 5 q^{27} + 10 q^{29} - 8 q^{31} + 13 q^{33} - 12 q^{37} - 24 q^{39} + 3 q^{41} - 14 q^{43} + 2 q^{47} + 8 q^{49} + 7 q^{51} + 4 q^{53} + 10 q^{57} + 20 q^{59} + 18 q^{61} - 4 q^{63} - 23 q^{67} - 16 q^{69} - 8 q^{71} - q^{73} + 26 q^{77} - 10 q^{79} - 16 q^{81} - 14 q^{83} - 20 q^{87} + 5 q^{89} - 48 q^{91} - 22 q^{93} - 2 q^{97} + q^{99}+O(q^{100})$$ 4 * q + q^3 + 2 * q^7 - 3 * q^9 + 2 * q^11 - 6 * q^13 - 7 * q^17 - 15 * q^19 + 18 * q^21 + 6 * q^23 - 5 * q^27 + 10 * q^29 - 8 * q^31 + 13 * q^33 - 12 * q^37 - 24 * q^39 + 3 * q^41 - 14 * q^43 + 2 * q^47 + 8 * q^49 + 7 * q^51 + 4 * q^53 + 10 * q^57 + 20 * q^59 + 18 * q^61 - 4 * q^63 - 23 * q^67 - 16 * q^69 - 8 * q^71 - q^73 + 26 * q^77 - 10 * q^79 - 16 * q^81 - 14 * q^83 - 20 * q^87 + 5 * q^89 - 48 * q^91 - 22 * q^93 - 2 * q^97 + q^99

Basis of coefficient ring in terms of $$\nu = \zeta_{15} + \zeta_{15}^{-1}$$:

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

## 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
 1.82709 −1.95630 −0.209057 1.33826
0 −1.95630 0 0 0 −3.91259 0 0.827091 0
1.2 0 −0.209057 0 0 0 −0.418114 0 −2.95630 0
1.3 0 1.33826 0 0 0 2.67652 0 −1.20906 0
1.4 0 1.82709 0 0 0 3.65418 0 0.338261 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
$$2$$ $$-1$$
$$5$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 10000.2.a.w 4
4.b odd 2 1 1250.2.a.e 4
5.b even 2 1 10000.2.a.s 4
20.d odd 2 1 1250.2.a.k yes 4
20.e even 4 2 1250.2.b.f 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1250.2.a.e 4 4.b odd 2 1
1250.2.a.k yes 4 20.d odd 2 1
1250.2.b.f 8 20.e even 4 2
10000.2.a.s 4 5.b even 2 1
10000.2.a.w 4 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(10000))$$:

 $$T_{3}^{4} - T_{3}^{3} - 4T_{3}^{2} + 4T_{3} + 1$$ T3^4 - T3^3 - 4*T3^2 + 4*T3 + 1 $$T_{7}^{4} - 2T_{7}^{3} - 16T_{7}^{2} + 32T_{7} + 16$$ T7^4 - 2*T7^3 - 16*T7^2 + 32*T7 + 16 $$T_{11}^{4} - 2T_{11}^{3} - 16T_{11}^{2} + 2T_{11} + 31$$ T11^4 - 2*T11^3 - 16*T11^2 + 2*T11 + 31

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} - T^{3} - 4 T^{2} + 4 T + 1$$
$5$ $$T^{4}$$
$7$ $$T^{4} - 2 T^{3} - 16 T^{2} + 32 T + 16$$
$11$ $$T^{4} - 2 T^{3} - 16 T^{2} + 2 T + 31$$
$13$ $$T^{4} + 6 T^{3} - 24 T^{2} - 144 T - 144$$
$17$ $$T^{4} + 7 T^{3} - 16 T^{2} - 112 T + 61$$
$19$ $$T^{4} + 15 T^{3} + 50 T^{2} + \cdots - 755$$
$23$ $$T^{4} - 6 T^{3} - 64 T^{2} + 384 T - 464$$
$29$ $$T^{4} - 10 T^{3} - 60 T^{2} + \cdots - 2480$$
$31$ $$T^{4} + 8 T^{3} - 16 T^{2} - 248 T - 464$$
$37$ $$T^{4} + 12 T^{3} - 16 T^{2} + \cdots - 1424$$
$41$ $$T^{4} - 3 T^{3} - 76 T^{2} + 408 T - 359$$
$43$ $$T^{4} + 14 T^{3} + 21 T^{2} + \cdots - 1289$$
$47$ $$T^{4} - 2 T^{3} - 76 T^{2} - 88 T + 16$$
$53$ $$T^{4} - 4 T^{3} - 64 T^{2} + 256 T + 256$$
$59$ $$T^{4} - 20 T^{3} + 95 T^{2} + \cdots - 905$$
$61$ $$T^{4} - 18 T^{3} - 76 T^{2} + \cdots - 9584$$
$67$ $$T^{4} + 23 T^{3} + 134 T^{2} + \cdots - 2039$$
$71$ $$T^{4} + 8 T^{3} - 76 T^{2} - 128 T + 16$$
$73$ $$T^{4} + T^{3} - 124 T^{2} + 476 T - 59$$
$79$ $$T^{4} + 10 T^{3} - 60 T^{2} + \cdots - 2480$$
$83$ $$T^{4} + 14 T^{3} - 64 T^{2} + \cdots - 3089$$
$89$ $$T^{4} - 5 T^{3} - 160 T^{2} - 100 T + 25$$
$97$ $$(T^{2} + T - 101)^{2}$$