# Properties

 Label 1815.2.a.y Level $1815$ Weight $2$ Character orbit 1815.a Self dual yes Analytic conductor $14.493$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1815 = 3 \cdot 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1815.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$14.4928479669$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.6.437199552.1 Defining polynomial: $$x^{6} - 13 x^{4} + 49 x^{2} - 48$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} - q^{3} + ( 2 + \beta_{2} ) q^{4} + q^{5} -\beta_{1} q^{6} + ( \beta_{1} - \beta_{3} - \beta_{4} ) q^{7} + ( \beta_{1} + \beta_{3} ) q^{8} + q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} - q^{3} + ( 2 + \beta_{2} ) q^{4} + q^{5} -\beta_{1} q^{6} + ( \beta_{1} - \beta_{3} - \beta_{4} ) q^{7} + ( \beta_{1} + \beta_{3} ) q^{8} + q^{9} + \beta_{1} q^{10} + ( -2 - \beta_{2} ) q^{12} + 2 \beta_{1} q^{13} + ( 2 - 2 \beta_{2} - 2 \beta_{5} ) q^{14} - q^{15} + ( 1 + 2 \beta_{2} + \beta_{5} ) q^{16} + ( \beta_{1} + \beta_{3} - 2 \beta_{4} ) q^{17} + \beta_{1} q^{18} + ( -\beta_{1} - \beta_{3} + 3 \beta_{4} ) q^{19} + ( 2 + \beta_{2} ) q^{20} + ( -\beta_{1} + \beta_{3} + \beta_{4} ) q^{21} + ( -1 + \beta_{2} + \beta_{5} ) q^{23} + ( -\beta_{1} - \beta_{3} ) q^{24} + q^{25} + ( 8 + 2 \beta_{2} ) q^{26} - q^{27} + ( -2 \beta_{3} - 6 \beta_{4} ) q^{28} + 2 \beta_{1} q^{29} -\beta_{1} q^{30} + ( 3 + 2 \beta_{5} ) q^{31} + ( \beta_{3} + 4 \beta_{4} ) q^{32} + ( 3 + 4 \beta_{2} - \beta_{5} ) q^{34} + ( \beta_{1} - \beta_{3} - \beta_{4} ) q^{35} + ( 2 + \beta_{2} ) q^{36} + ( 2 - \beta_{2} - \beta_{5} ) q^{37} + ( -2 - 4 \beta_{2} + 2 \beta_{5} ) q^{38} -2 \beta_{1} q^{39} + ( \beta_{1} + \beta_{3} ) q^{40} -2 \beta_{4} q^{41} + ( -2 + 2 \beta_{2} + 2 \beta_{5} ) q^{42} + ( -2 \beta_{3} + 2 \beta_{4} ) q^{43} + q^{45} + ( -\beta_{1} + 2 \beta_{3} + 4 \beta_{4} ) q^{46} + ( -1 + 3 \beta_{2} - \beta_{5} ) q^{47} + ( -1 - 2 \beta_{2} - \beta_{5} ) q^{48} + ( 11 + \beta_{2} - \beta_{5} ) q^{49} + \beta_{1} q^{50} + ( -\beta_{1} - \beta_{3} + 2 \beta_{4} ) q^{51} + ( 6 \beta_{1} + 2 \beta_{3} ) q^{52} + ( -3 + \beta_{2} - \beta_{5} ) q^{53} -\beta_{1} q^{54} + ( -12 - 2 \beta_{2} - 4 \beta_{5} ) q^{56} + ( \beta_{1} + \beta_{3} - 3 \beta_{4} ) q^{57} + ( 8 + 2 \beta_{2} ) q^{58} + ( -2 - 4 \beta_{2} + 2 \beta_{5} ) q^{59} + ( -2 - \beta_{2} ) q^{60} + ( -4 \beta_{1} - \beta_{4} ) q^{61} + ( \beta_{1} + 2 \beta_{3} + 8 \beta_{4} ) q^{62} + ( \beta_{1} - \beta_{3} - \beta_{4} ) q^{63} + ( 3 - \beta_{2} + 3 \beta_{5} ) q^{64} + 2 \beta_{1} q^{65} + ( 2 - \beta_{2} - \beta_{5} ) q^{67} + ( 6 \beta_{1} + \beta_{3} ) q^{68} + ( 1 - \beta_{2} - \beta_{5} ) q^{69} + ( 2 - 2 \beta_{2} - 2 \beta_{5} ) q^{70} + ( -2 \beta_{2} + 2 \beta_{5} ) q^{71} + ( \beta_{1} + \beta_{3} ) q^{72} + ( 3 \beta_{1} - \beta_{3} + \beta_{4} ) q^{73} + ( 2 \beta_{1} - 2 \beta_{3} - 4 \beta_{4} ) q^{74} - q^{75} + ( -6 \beta_{1} + 2 \beta_{4} ) q^{76} + ( -8 - 2 \beta_{2} ) q^{78} + ( 2 \beta_{1} - 7 \beta_{4} ) q^{79} + ( 1 + 2 \beta_{2} + \beta_{5} ) q^{80} + q^{81} + ( -2 - 2 \beta_{5} ) q^{82} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{83} + ( 2 \beta_{3} + 6 \beta_{4} ) q^{84} + ( \beta_{1} + \beta_{3} - 2 \beta_{4} ) q^{85} -6 \beta_{2} q^{86} -2 \beta_{1} q^{87} + ( 8 + 2 \beta_{5} ) q^{89} + \beta_{1} q^{90} + ( 4 - 4 \beta_{2} - 4 \beta_{5} ) q^{91} + ( 4 + 3 \beta_{2} + 4 \beta_{5} ) q^{92} + ( -3 - 2 \beta_{5} ) q^{93} + ( 3 \beta_{1} + 2 \beta_{3} - 4 \beta_{4} ) q^{94} + ( -\beta_{1} - \beta_{3} + 3 \beta_{4} ) q^{95} + ( -\beta_{3} - 4 \beta_{4} ) q^{96} + ( 2 - 5 \beta_{2} - \beta_{5} ) q^{97} + ( 13 \beta_{1} - 4 \beta_{4} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q - 6q^{3} + 14q^{4} + 6q^{5} + 6q^{9} + O(q^{10})$$ $$6q - 6q^{3} + 14q^{4} + 6q^{5} + 6q^{9} - 14q^{12} + 8q^{14} - 6q^{15} + 10q^{16} + 14q^{20} - 4q^{23} + 6q^{25} + 52q^{26} - 6q^{27} + 18q^{31} + 26q^{34} + 14q^{36} + 10q^{37} - 20q^{38} - 8q^{42} + 6q^{45} - 10q^{48} + 68q^{49} - 16q^{53} - 76q^{56} + 52q^{58} - 20q^{59} - 14q^{60} + 16q^{64} + 10q^{67} + 4q^{69} + 8q^{70} - 4q^{71} - 6q^{75} - 52q^{78} + 10q^{80} + 6q^{81} - 12q^{82} - 12q^{86} + 48q^{89} + 16q^{91} + 30q^{92} - 18q^{93} + 2q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 13 x^{4} + 49 x^{2} - 48$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 4$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - 5 \nu$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{5} - 9 \nu^{3} + 17 \nu$$$$)/4$$ $$\beta_{5}$$ $$=$$ $$\nu^{4} - 8 \nu^{2} + 11$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 4$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + 5 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{5} + 8 \beta_{2} + 21$$ $$\nu^{5}$$ $$=$$ $$4 \beta_{4} + 9 \beta_{3} + 28 \beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −2.63162 −2.13353 −1.23396 1.23396 2.13353 2.63162
−2.63162 −1.00000 4.92542 1.00000 2.63162 4.16741 −7.69860 1.00000 −2.63162
1.2 −2.13353 −1.00000 2.55193 1.00000 2.13353 −4.82155 −1.17756 1.00000 −2.13353
1.3 −1.23396 −1.00000 −0.477352 1.00000 1.23396 −3.79281 3.05694 1.00000 −1.23396
1.4 1.23396 −1.00000 −0.477352 1.00000 −1.23396 3.79281 −3.05694 1.00000 1.23396
1.5 2.13353 −1.00000 2.55193 1.00000 −2.13353 4.82155 1.17756 1.00000 2.13353
1.6 2.63162 −1.00000 4.92542 1.00000 −2.63162 −4.16741 7.69860 1.00000 2.63162
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$
$$5$$ $$-1$$
$$11$$ $$1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1815.2.a.y 6
3.b odd 2 1 5445.2.a.bz 6
5.b even 2 1 9075.2.a.dq 6
11.b odd 2 1 inner 1815.2.a.y 6
33.d even 2 1 5445.2.a.bz 6
55.d odd 2 1 9075.2.a.dq 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1815.2.a.y 6 1.a even 1 1 trivial
1815.2.a.y 6 11.b odd 2 1 inner
5445.2.a.bz 6 3.b odd 2 1
5445.2.a.bz 6 33.d even 2 1
9075.2.a.dq 6 5.b even 2 1
9075.2.a.dq 6 55.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1815))$$:

 $$T_{2}^{6} - 13 T_{2}^{4} + 49 T_{2}^{2} - 48$$ $$T_{7}^{6} - 55 T_{7}^{4} + 988 T_{7}^{2} - 5808$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-48 + 49 T^{2} - 13 T^{4} + T^{6}$$
$3$ $$( 1 + T )^{6}$$
$5$ $$( -1 + T )^{6}$$
$7$ $$-5808 + 988 T^{2} - 55 T^{4} + T^{6}$$
$11$ $$T^{6}$$
$13$ $$-3072 + 784 T^{2} - 52 T^{4} + T^{6}$$
$17$ $$-16428 + 2065 T^{2} - 82 T^{4} + T^{6}$$
$19$ $$-17328 + 3448 T^{2} - 115 T^{4} + T^{6}$$
$23$ $$( -66 - 29 T + 2 T^{2} + T^{3} )^{2}$$
$29$ $$-3072 + 784 T^{2} - 52 T^{4} + T^{6}$$
$31$ $$( 341 - 45 T - 9 T^{2} + T^{3} )^{2}$$
$37$ $$( 92 - 22 T - 5 T^{2} + T^{3} )^{2}$$
$41$ $$( -12 + T^{2} )^{3}$$
$43$ $$-15552 + 6912 T^{2} - 192 T^{4} + T^{6}$$
$47$ $$( 216 - 69 T + T^{3} )^{2}$$
$53$ $$( -54 + T + 8 T^{2} + T^{3} )^{2}$$
$59$ $$( -912 - 116 T + 10 T^{2} + T^{3} )^{2}$$
$61$ $$-309123 + 15691 T^{2} - 241 T^{4} + T^{6}$$
$67$ $$( 92 - 22 T - 5 T^{2} + T^{3} )^{2}$$
$71$ $$( 96 - 80 T + 2 T^{2} + T^{3} )^{2}$$
$73$ $$-62208 + 5616 T^{2} - 147 T^{4} + T^{6}$$
$79$ $$-1179387 + 42091 T^{2} - 409 T^{4} + T^{6}$$
$83$ $$-110592 + 8464 T^{2} - 184 T^{4} + T^{6}$$
$89$ $$( 216 + 120 T - 24 T^{2} + T^{3} )^{2}$$
$97$ $$( 844 - 226 T - T^{2} + T^{3} )^{2}$$