# Properties

 Label 1815.2.a.o Level $1815$ Weight $2$ Character orbit 1815.a Self dual yes Analytic conductor $14.493$ Analytic rank $1$ Dimension $4$ CM no Inner twists $1$

# 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: $$1$$ Dimension: $$4$$ Coefficient field: 4.4.725.1 Defining polynomial: $$x^{4} - x^{3} - 3 x^{2} + x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 165) 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.

 $$f(q)$$ $$=$$ $$q + ( -2 + \beta_{1} + \beta_{2} ) q^{2} + q^{3} + ( 3 - \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{4} - q^{5} + ( -2 + \beta_{1} + \beta_{2} ) q^{6} + ( -2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{7} + ( -4 + \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{8} + q^{9} +O(q^{10})$$ $$q + ( -2 + \beta_{1} + \beta_{2} ) q^{2} + q^{3} + ( 3 - \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{4} - q^{5} + ( -2 + \beta_{1} + \beta_{2} ) q^{6} + ( -2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{7} + ( -4 + \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{8} + q^{9} + ( 2 - \beta_{1} - \beta_{2} ) q^{10} + ( 3 - \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{12} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{13} + ( 1 + 3 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} ) q^{14} - q^{15} + ( 5 - 5 \beta_{1} - 5 \beta_{2} + 5 \beta_{3} ) q^{16} -5 q^{17} + ( -2 + \beta_{1} + \beta_{2} ) q^{18} + ( 1 + 3 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} ) q^{19} + ( -3 + \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{20} + ( -2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{21} + ( -\beta_{1} - \beta_{2} - \beta_{3} ) q^{23} + ( -4 + \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{24} + q^{25} + ( 1 + 2 \beta_{2} - \beta_{3} ) q^{26} + q^{27} + ( -4 - 5 \beta_{1} + 9 \beta_{3} ) q^{28} + ( 1 - \beta_{1} - 4 \beta_{2} ) q^{29} + ( 2 - \beta_{1} - \beta_{2} ) q^{30} + ( 3 \beta_{1} - 2 \beta_{3} ) q^{31} + ( -7 + 8 \beta_{1} + 4 \beta_{2} - 9 \beta_{3} ) q^{32} + ( 10 - 5 \beta_{1} - 5 \beta_{2} ) q^{34} + ( 2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{35} + ( 3 - \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{36} + ( \beta_{1} + \beta_{2} - 5 \beta_{3} ) q^{37} + ( -4 - 7 \beta_{1} + 3 \beta_{2} + 9 \beta_{3} ) q^{38} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{39} + ( 4 - \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{40} + ( -6 - 4 \beta_{1} - \beta_{2} + 5 \beta_{3} ) q^{41} + ( 1 + 3 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} ) q^{42} + ( 3 + \beta_{2} - 6 \beta_{3} ) q^{43} - q^{45} + ( -1 - 3 \beta_{1} - \beta_{2} + \beta_{3} ) q^{46} + ( -6 - 4 \beta_{1} + \beta_{2} + 3 \beta_{3} ) q^{47} + ( 5 - 5 \beta_{1} - 5 \beta_{2} + 5 \beta_{3} ) q^{48} + ( 4 - 2 \beta_{1} + \beta_{2} - 4 \beta_{3} ) q^{49} + ( -2 + \beta_{1} + \beta_{2} ) q^{50} -5 q^{51} + ( 2 - \beta_{1} ) q^{52} + ( 1 + 6 \beta_{1} + 3 \beta_{2} - 5 \beta_{3} ) q^{53} + ( -2 + \beta_{1} + \beta_{2} ) q^{54} + ( 6 + 8 \beta_{1} + 4 \beta_{2} - 15 \beta_{3} ) q^{56} + ( 1 + 3 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} ) q^{57} + ( -6 - 3 \beta_{1} + 4 \beta_{2} - \beta_{3} ) q^{58} + ( -5 + \beta_{1} + \beta_{2} + 6 \beta_{3} ) q^{59} + ( -3 + \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{60} + ( -3 + 3 \beta_{1} + 6 \beta_{2} + 2 \beta_{3} ) q^{61} + ( -4 \beta_{1} + \beta_{2} + 7 \beta_{3} ) q^{62} + ( -2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{63} + ( 8 - 11 \beta_{1} - 2 \beta_{2} + 16 \beta_{3} ) q^{64} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{65} + ( -5 - 5 \beta_{1} + 5 \beta_{2} + \beta_{3} ) q^{67} + ( -15 + 5 \beta_{1} + 10 \beta_{2} - 5 \beta_{3} ) q^{68} + ( -\beta_{1} - \beta_{2} - \beta_{3} ) q^{69} + ( -1 - 3 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} ) q^{70} + ( -4 + 3 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} ) q^{71} + ( -4 + \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{72} + ( -2 + 3 \beta_{1} - 5 \beta_{2} - 4 \beta_{3} ) q^{73} + ( 1 - 9 \beta_{1} - 5 \beta_{2} + 11 \beta_{3} ) q^{74} + q^{75} + ( 9 + 11 \beta_{1} - \beta_{2} - 19 \beta_{3} ) q^{76} + ( 1 + 2 \beta_{2} - \beta_{3} ) q^{78} + ( -2 - 2 \beta_{1} - 2 \beta_{2} + 7 \beta_{3} ) q^{79} + ( -5 + 5 \beta_{1} + 5 \beta_{2} - 5 \beta_{3} ) q^{80} + q^{81} + ( 11 + 3 \beta_{1} - 4 \beta_{2} - 14 \beta_{3} ) q^{82} + ( -10 - 5 \beta_{1} - 2 \beta_{2} + 8 \beta_{3} ) q^{83} + ( -4 - 5 \beta_{1} + 9 \beta_{3} ) q^{84} + 5 q^{85} + ( -5 - 8 \beta_{1} - 4 \beta_{2} + 12 \beta_{3} ) q^{86} + ( 1 - \beta_{1} - 4 \beta_{2} ) q^{87} + ( 2 + 2 \beta_{1} + 3 \beta_{3} ) q^{89} + ( 2 - \beta_{1} - \beta_{2} ) q^{90} + ( -6 + 2 \beta_{1} - 5 \beta_{2} + 4 \beta_{3} ) q^{91} + ( 1 + 2 \beta_{1} - 3 \beta_{3} ) q^{92} + ( 3 \beta_{1} - 2 \beta_{3} ) q^{93} + ( 13 + \beta_{1} - 8 \beta_{2} - 10 \beta_{3} ) q^{94} + ( -1 - 3 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} ) q^{95} + ( -7 + 8 \beta_{1} + 4 \beta_{2} - 9 \beta_{3} ) q^{96} + ( -1 + 4 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{97} + ( -7 - 3 \beta_{1} - 3 \beta_{2} + 6 \beta_{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 5 q^{2} + 4 q^{3} + 9 q^{4} - 4 q^{5} - 5 q^{6} + 2 q^{7} - 15 q^{8} + 4 q^{9} + O(q^{10})$$ $$4 q - 5 q^{2} + 4 q^{3} + 9 q^{4} - 4 q^{5} - 5 q^{6} + 2 q^{7} - 15 q^{8} + 4 q^{9} + 5 q^{10} + 9 q^{12} - 3 q^{13} - 5 q^{14} - 4 q^{15} + 15 q^{16} - 20 q^{17} - 5 q^{18} - 3 q^{19} - 9 q^{20} + 2 q^{21} - 5 q^{23} - 15 q^{24} + 4 q^{25} + 6 q^{26} + 4 q^{27} - 3 q^{28} - 5 q^{29} + 5 q^{30} - q^{31} - 30 q^{32} + 25 q^{34} - 2 q^{35} + 9 q^{36} - 7 q^{37} + q^{38} - 3 q^{39} + 15 q^{40} - 20 q^{41} - 5 q^{42} + 2 q^{43} - 4 q^{45} - 7 q^{46} - 20 q^{47} + 15 q^{48} + 8 q^{49} - 5 q^{50} - 20 q^{51} + 7 q^{52} + 6 q^{53} - 5 q^{54} + 10 q^{56} - 3 q^{57} - 21 q^{58} - 5 q^{59} - 9 q^{60} + 7 q^{61} + 12 q^{62} + 2 q^{63} + 49 q^{64} + 3 q^{65} - 13 q^{67} - 45 q^{68} - 5 q^{69} + 5 q^{70} - 25 q^{71} - 15 q^{72} - 23 q^{73} + 7 q^{74} + 4 q^{75} + 7 q^{76} + 6 q^{78} - 15 q^{80} + 4 q^{81} + 11 q^{82} - 33 q^{83} - 3 q^{84} + 20 q^{85} - 12 q^{86} - 5 q^{87} + 16 q^{89} + 5 q^{90} - 24 q^{91} - q^{93} + 17 q^{94} + 3 q^{95} - 30 q^{96} - 25 q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 3 x^{2} + x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 1$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - \nu^{2} - 2 \nu + 1$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta_{1} + 1$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + \beta_{2} + 3 \beta_{1}$$

## 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
 −0.477260 0.737640 −1.35567 2.09529
−2.77222 1.00000 5.68522 −1.00000 −2.77222 2.27759 −10.2163 1.00000 2.77222
1.2 −2.45589 1.00000 4.03138 −1.00000 −2.45589 −3.28684 −4.98884 1.00000 2.45589
1.3 −1.16215 1.00000 −0.649414 −1.00000 −1.16215 4.28684 3.07901 1.00000 1.16215
1.4 1.39026 1.00000 −0.0671858 −1.00000 1.39026 −1.27759 −2.87392 1.00000 −1.39026
 $$n$$: 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

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 1815.2.a.o 4
3.b odd 2 1 5445.2.a.bv 4
5.b even 2 1 9075.2.a.dj 4
11.b odd 2 1 1815.2.a.x 4
11.d odd 10 2 165.2.m.a 8
33.d even 2 1 5445.2.a.be 4
33.f even 10 2 495.2.n.d 8
55.d odd 2 1 9075.2.a.cl 4
55.h odd 10 2 825.2.n.k 8
55.l even 20 4 825.2.bx.h 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.m.a 8 11.d odd 10 2
495.2.n.d 8 33.f even 10 2
825.2.n.k 8 55.h odd 10 2
825.2.bx.h 16 55.l even 20 4
1815.2.a.o 4 1.a even 1 1 trivial
1815.2.a.x 4 11.b odd 2 1
5445.2.a.be 4 33.d even 2 1
5445.2.a.bv 4 3.b odd 2 1
9075.2.a.cl 4 55.d odd 2 1
9075.2.a.dj 4 5.b even 2 1

## Hecke kernels

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

 $$T_{2}^{4} + 5 T_{2}^{3} + 4 T_{2}^{2} - 10 T_{2} - 11$$ $$T_{7}^{4} - 2 T_{7}^{3} - 16 T_{7}^{2} + 17 T_{7} + 41$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-11 - 10 T + 4 T^{2} + 5 T^{3} + T^{4}$$
$3$ $$( -1 + T )^{4}$$
$5$ $$( 1 + T )^{4}$$
$7$ $$41 + 17 T - 16 T^{2} - 2 T^{3} + T^{4}$$
$11$ $$T^{4}$$
$13$ $$-1 + 6 T - 10 T^{2} + 3 T^{3} + T^{4}$$
$17$ $$( 5 + T )^{4}$$
$19$ $$-31 - 204 T - 50 T^{2} + 3 T^{3} + T^{4}$$
$23$ $$-1 - 5 T - T^{2} + 5 T^{3} + T^{4}$$
$29$ $$539 - 140 T - 44 T^{2} + 5 T^{3} + T^{4}$$
$31$ $$139 - 7 T - 25 T^{2} + T^{3} + T^{4}$$
$37$ $$431 - 133 T - 37 T^{2} + 7 T^{3} + T^{4}$$
$41$ $$-2071 - 335 T + 86 T^{2} + 20 T^{3} + T^{4}$$
$43$ $$1861 + 63 T - 92 T^{2} - 2 T^{3} + T^{4}$$
$47$ $$-11 - 25 T + 94 T^{2} + 20 T^{3} + T^{4}$$
$53$ $$-1271 + 777 T - 100 T^{2} - 6 T^{3} + T^{4}$$
$59$ $$2299 - 325 T - 101 T^{2} + 5 T^{3} + T^{4}$$
$61$ $$1891 + 322 T - 136 T^{2} - 7 T^{3} + T^{4}$$
$67$ $$-3379 - 1768 T - 136 T^{2} + 13 T^{3} + T^{4}$$
$71$ $$-2351 - 25 T + 171 T^{2} + 25 T^{3} + T^{4}$$
$73$ $$-9199 - 1722 T + 48 T^{2} + 23 T^{3} + T^{4}$$
$79$ $$1199 - 210 T - 109 T^{2} + T^{4}$$
$83$ $$-12221 - 869 T + 265 T^{2} + 33 T^{3} + T^{4}$$
$89$ $$-271 + 132 T + 45 T^{2} - 16 T^{3} + T^{4}$$
$97$ $$-25 + 125 T - 60 T^{2} + T^{4}$$