# Properties

 Label 1815.4.a.k Level $1815$ Weight $4$ Character orbit 1815.a Self dual yes Analytic conductor $107.088$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1815 = 3 \cdot 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1815.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$107.088466660$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ 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 $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 \beta q^{2} + 3 q^{3} + 4 q^{4} -5 q^{5} + 6 \beta q^{6} + 16 \beta q^{7} -8 \beta q^{8} + 9 q^{9} +O(q^{10})$$ $$q + 2 \beta q^{2} + 3 q^{3} + 4 q^{4} -5 q^{5} + 6 \beta q^{6} + 16 \beta q^{7} -8 \beta q^{8} + 9 q^{9} -10 \beta q^{10} + 12 q^{12} -34 \beta q^{13} + 96 q^{14} -15 q^{15} -80 q^{16} + 11 \beta q^{17} + 18 \beta q^{18} -26 \beta q^{19} -20 q^{20} + 48 \beta q^{21} -75 q^{23} -24 \beta q^{24} + 25 q^{25} -204 q^{26} + 27 q^{27} + 64 \beta q^{28} -74 \beta q^{29} -30 \beta q^{30} -263 q^{31} -96 \beta q^{32} + 66 q^{34} -80 \beta q^{35} + 36 q^{36} -308 q^{37} -156 q^{38} -102 \beta q^{39} + 40 \beta q^{40} + 94 \beta q^{41} + 288 q^{42} -22 \beta q^{43} -45 q^{45} -150 \beta q^{46} + 93 q^{47} -240 q^{48} + 425 q^{49} + 50 \beta q^{50} + 33 \beta q^{51} -136 \beta q^{52} + 525 q^{53} + 54 \beta q^{54} -384 q^{56} -78 \beta q^{57} -444 q^{58} + 498 q^{59} -60 q^{60} -255 \beta q^{61} -526 \beta q^{62} + 144 \beta q^{63} + 64 q^{64} + 170 \beta q^{65} + 316 q^{67} + 44 \beta q^{68} -225 q^{69} -480 q^{70} -288 q^{71} -72 \beta q^{72} -536 \beta q^{73} -616 \beta q^{74} + 75 q^{75} -104 \beta q^{76} -612 q^{78} -657 \beta q^{79} + 400 q^{80} + 81 q^{81} + 564 q^{82} + 330 \beta q^{83} + 192 \beta q^{84} -55 \beta q^{85} -132 q^{86} -222 \beta q^{87} -180 q^{89} -90 \beta q^{90} -1632 q^{91} -300 q^{92} -789 q^{93} + 186 \beta q^{94} + 130 \beta q^{95} -288 \beta q^{96} -904 q^{97} + 850 \beta q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 6q^{3} + 8q^{4} - 10q^{5} + 18q^{9} + O(q^{10})$$ $$2q + 6q^{3} + 8q^{4} - 10q^{5} + 18q^{9} + 24q^{12} + 192q^{14} - 30q^{15} - 160q^{16} - 40q^{20} - 150q^{23} + 50q^{25} - 408q^{26} + 54q^{27} - 526q^{31} + 132q^{34} + 72q^{36} - 616q^{37} - 312q^{38} + 576q^{42} - 90q^{45} + 186q^{47} - 480q^{48} + 850q^{49} + 1050q^{53} - 768q^{56} - 888q^{58} + 996q^{59} - 120q^{60} + 128q^{64} + 632q^{67} - 450q^{69} - 960q^{70} - 576q^{71} + 150q^{75} - 1224q^{78} + 800q^{80} + 162q^{81} + 1128q^{82} - 264q^{86} - 360q^{89} - 3264q^{91} - 600q^{92} - 1578q^{93} - 1808q^{97} + O(q^{100})$$

## 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
 −1.73205 1.73205
−3.46410 3.00000 4.00000 −5.00000 −10.3923 −27.7128 13.8564 9.00000 17.3205
1.2 3.46410 3.00000 4.00000 −5.00000 10.3923 27.7128 −13.8564 9.00000 −17.3205
 $$n$$: e.g. 2-40 or 990-1000 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 1815.4.a.k 2
11.b odd 2 1 inner 1815.4.a.k 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1815.4.a.k 2 1.a even 1 1 trivial
1815.4.a.k 2 11.b 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(1815))$$:

 $$T_{2}^{2} - 12$$ $$T_{7}^{2} - 768$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-12 + T^{2}$$
$3$ $$( -3 + T )^{2}$$
$5$ $$( 5 + T )^{2}$$
$7$ $$-768 + T^{2}$$
$11$ $$T^{2}$$
$13$ $$-3468 + T^{2}$$
$17$ $$-363 + T^{2}$$
$19$ $$-2028 + T^{2}$$
$23$ $$( 75 + T )^{2}$$
$29$ $$-16428 + T^{2}$$
$31$ $$( 263 + T )^{2}$$
$37$ $$( 308 + T )^{2}$$
$41$ $$-26508 + T^{2}$$
$43$ $$-1452 + T^{2}$$
$47$ $$( -93 + T )^{2}$$
$53$ $$( -525 + T )^{2}$$
$59$ $$( -498 + T )^{2}$$
$61$ $$-195075 + T^{2}$$
$67$ $$( -316 + T )^{2}$$
$71$ $$( 288 + T )^{2}$$
$73$ $$-861888 + T^{2}$$
$79$ $$-1294947 + T^{2}$$
$83$ $$-326700 + T^{2}$$
$89$ $$( 180 + T )^{2}$$
$97$ $$( 904 + T )^{2}$$