# Properties

 Label 1815.2.a.u Level $1815$ Weight $2$ Character orbit 1815.a Self dual yes Analytic conductor $14.493$ Analytic rank $0$ 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: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{15})^+$$ Defining polynomial: $$x^{4} - x^{3} - 4 x^{2} + 4 x + 1$$ 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.

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

## 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
 1.82709 −1.95630 −0.209057 1.33826
−1.95630 −1.00000 1.82709 1.00000 1.95630 1.54732 0.338261 1.00000 −1.95630
1.2 −0.209057 −1.00000 −1.95630 1.00000 0.209057 0.488830 0.827091 1.00000 −0.209057
1.3 1.33826 −1.00000 −0.209057 1.00000 −1.33826 −3.78339 −2.95630 1.00000 1.33826
1.4 1.82709 −1.00000 1.33826 1.00000 −1.82709 1.74724 −1.20906 1.00000 1.82709
 $$n$$: 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.u 4
3.b odd 2 1 5445.2.a.bj 4
5.b even 2 1 9075.2.a.co 4
11.b odd 2 1 1815.2.a.q 4
11.c even 5 2 165.2.m.c 8
33.d even 2 1 5445.2.a.bq 4
33.h odd 10 2 495.2.n.c 8
55.d odd 2 1 9075.2.a.df 4
55.j even 10 2 825.2.n.j 8
55.k odd 20 4 825.2.bx.e 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.m.c 8 11.c even 5 2
495.2.n.c 8 33.h odd 10 2
825.2.n.j 8 55.j even 10 2
825.2.bx.e 16 55.k odd 20 4
1815.2.a.q 4 11.b odd 2 1
1815.2.a.u 4 1.a even 1 1 trivial
5445.2.a.bj 4 3.b odd 2 1
5445.2.a.bq 4 33.d even 2 1
9075.2.a.co 4 5.b even 2 1
9075.2.a.df 4 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1815))$$:

 $$T_{2}^{4} - T_{2}^{3} - 4 T_{2}^{2} + 4 T_{2} + 1$$ $$T_{7}^{4} - 10 T_{7}^{2} + 15 T_{7} - 5$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 4 T - 4 T^{2} - T^{3} + T^{4}$$
$3$ $$( 1 + T )^{4}$$
$5$ $$( -1 + T )^{4}$$
$7$ $$-5 + 15 T - 10 T^{2} + T^{4}$$
$11$ $$T^{4}$$
$13$ $$-359 + 202 T - 16 T^{2} - 7 T^{3} + T^{4}$$
$17$ $$-59 + 88 T - 6 T^{2} - 8 T^{3} + T^{4}$$
$19$ $$-239 + 184 T + 6 T^{2} - 11 T^{3} + T^{4}$$
$23$ $$-5 - 25 T - 25 T^{2} - 5 T^{3} + T^{4}$$
$29$ $$-419 - 38 T + 84 T^{2} - 17 T^{3} + T^{4}$$
$31$ $$-5 - 235 T - 45 T^{2} + 5 T^{3} + T^{4}$$
$37$ $$-155 - 45 T + 65 T^{2} - 15 T^{3} + T^{4}$$
$41$ $$-5 + 145 T - 10 T^{2} - 10 T^{3} + T^{4}$$
$43$ $$-569 - 541 T - 124 T^{2} + 4 T^{3} + T^{4}$$
$47$ $$121 - 143 T - 16 T^{2} + 8 T^{3} + T^{4}$$
$53$ $$25 + 25 T - 10 T^{2} - 10 T^{3} + T^{4}$$
$59$ $$1341 + 261 T - 69 T^{2} - 9 T^{3} + T^{4}$$
$61$ $$6571 - 2998 T + 504 T^{2} - 37 T^{3} + T^{4}$$
$67$ $$31 - 72 T - 46 T^{2} - 3 T^{3} + T^{4}$$
$71$ $$-389 + 83 T + 39 T^{2} - 13 T^{3} + T^{4}$$
$73$ $$-4205 + 1320 T - 40 T^{2} - 15 T^{3} + T^{4}$$
$79$ $$-305 - 10 T + 95 T^{2} - 20 T^{3} + T^{4}$$
$83$ $$-14669 - 5347 T - 261 T^{2} + 17 T^{3} + T^{4}$$
$89$ $$-31869 - 6066 T - 129 T^{2} + 24 T^{3} + T^{4}$$
$97$ $$61 + 463 T + 284 T^{2} + 32 T^{3} + T^{4}$$