# Properties

 Label 2001.2.a.h Level $2001$ Weight $2$ Character orbit 2001.a Self dual yes Analytic conductor $15.978$ Analytic rank $1$ Dimension $5$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2001 = 3 \cdot 23 \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2001.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$15.9780654445$$ Analytic rank: $$1$$ Dimension: $$5$$ Coefficient field: 5.5.312617.1 Defining polynomial: $$x^{5} - 2 x^{4} - 5 x^{3} + 11 x^{2} - x - 3$$ Coefficient ring: $$\Z[a_1, a_2]$$ 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 $$1,\beta_1,\beta_2,\beta_3,\beta_4$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{3} q^{2} - q^{3} + ( 2 - \beta_{1} ) q^{4} + ( -1 + \beta_{1} + \beta_{2} ) q^{5} + \beta_{3} q^{6} + ( -1 - \beta_{2} ) q^{7} + ( -1 + 2 \beta_{1} + \beta_{2} - \beta_{4} ) q^{8} + q^{9} +O(q^{10})$$ $$q -\beta_{3} q^{2} - q^{3} + ( 2 - \beta_{1} ) q^{4} + ( -1 + \beta_{1} + \beta_{2} ) q^{5} + \beta_{3} q^{6} + ( -1 - \beta_{2} ) q^{7} + ( -1 + 2 \beta_{1} + \beta_{2} - \beta_{4} ) q^{8} + q^{9} + ( 3 - 2 \beta_{1} - \beta_{4} ) q^{10} + ( -2 + \beta_{4} ) q^{11} + ( -2 + \beta_{1} ) q^{12} + ( 1 - \beta_{1} - \beta_{2} + \beta_{4} ) q^{13} + ( -2 - \beta_{2} + 2 \beta_{3} + 2 \beta_{4} ) q^{14} + ( 1 - \beta_{1} - \beta_{2} ) q^{15} + ( -2 - 2 \beta_{1} + \beta_{3} + \beta_{4} ) q^{16} + \beta_{1} q^{17} -\beta_{3} q^{18} + ( -1 - 2 \beta_{4} ) q^{19} + ( -2 + 2 \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{20} + ( 1 + \beta_{2} ) q^{21} + ( 2 - \beta_{2} + \beta_{3} - \beta_{4} ) q^{22} + q^{23} + ( 1 - 2 \beta_{1} - \beta_{2} + \beta_{4} ) q^{24} + \beta_{3} q^{25} + ( -1 + 2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{26} - q^{27} + ( -4 + 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{28} + q^{29} + ( -3 + 2 \beta_{1} + \beta_{4} ) q^{30} + ( -2 - \beta_{2} + 2 \beta_{3} ) q^{31} + ( -2 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} ) q^{32} + ( 2 - \beta_{4} ) q^{33} + ( 1 - 2 \beta_{1} - \beta_{2} + \beta_{4} ) q^{34} + ( -3 - 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} ) q^{35} + ( 2 - \beta_{1} ) q^{36} + ( 2 - 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} ) q^{37} + ( -4 + 2 \beta_{2} + 3 \beta_{3} + 2 \beta_{4} ) q^{38} + ( -1 + \beta_{1} + \beta_{2} - \beta_{4} ) q^{39} + ( 4 - 2 \beta_{1} + 2 \beta_{3} + 3 \beta_{4} ) q^{40} + ( -1 + 2 \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{41} + ( 2 + \beta_{2} - 2 \beta_{3} - 2 \beta_{4} ) q^{42} + ( -1 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} ) q^{43} + ( -4 + \beta_{1} + \beta_{4} ) q^{44} + ( -1 + \beta_{1} + \beta_{2} ) q^{45} -\beta_{3} q^{46} + ( -5 - \beta_{1} + 2 \beta_{2} + 4 \beta_{3} + 3 \beta_{4} ) q^{47} + ( 2 + 2 \beta_{1} - \beta_{3} - \beta_{4} ) q^{48} + ( 2 \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{49} + ( -4 + \beta_{1} ) q^{50} -\beta_{1} q^{51} + ( 2 - 3 \beta_{1} - \beta_{2} + 2 \beta_{3} + 2 \beta_{4} ) q^{52} + ( -1 + 2 \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} ) q^{53} + \beta_{3} q^{54} + ( 3 - \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{55} + ( -3 \beta_{1} - \beta_{2} + \beta_{3} ) q^{56} + ( 1 + 2 \beta_{4} ) q^{57} -\beta_{3} q^{58} + ( 2 + 4 \beta_{1} - 2 \beta_{3} - 4 \beta_{4} ) q^{59} + ( 2 - 2 \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} ) q^{60} + ( 3 + 3 \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} ) q^{61} + ( -10 + 2 \beta_{1} - \beta_{2} + 3 \beta_{3} + 2 \beta_{4} ) q^{62} + ( -1 - \beta_{2} ) q^{63} + ( -3 + 3 \beta_{1} - \beta_{2} + 2 \beta_{3} + 2 \beta_{4} ) q^{64} + ( -4 + 2 \beta_{1} + \beta_{2} - 2 \beta_{4} ) q^{65} + ( -2 + \beta_{2} - \beta_{3} + \beta_{4} ) q^{66} + ( -\beta_{1} - 4 \beta_{4} ) q^{67} + ( -2 + 2 \beta_{1} - \beta_{3} - \beta_{4} ) q^{68} - q^{69} + ( -6 + 5 \beta_{1} + 3 \beta_{3} - \beta_{4} ) q^{70} + ( -3 + \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} ) q^{71} + ( -1 + 2 \beta_{1} + \beta_{2} - \beta_{4} ) q^{72} + ( 7 - 3 \beta_{1} - 3 \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{73} + ( -2 + 5 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} - 5 \beta_{4} ) q^{74} -\beta_{3} q^{75} + ( -2 + 3 \beta_{1} - 2 \beta_{4} ) q^{76} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} ) q^{77} + ( 1 - 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{78} + ( 2 \beta_{1} + \beta_{2} + 2 \beta_{4} ) q^{79} + ( 2 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} ) q^{80} + q^{81} + ( 4 - \beta_{1} + 4 \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{82} + ( 5 - 7 \beta_{1} - 2 \beta_{3} ) q^{83} + ( 4 - 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{84} + ( \beta_{2} + 2 \beta_{3} + \beta_{4} ) q^{85} + ( -12 + 5 \beta_{1} + \beta_{2} + 4 \beta_{3} + 3 \beta_{4} ) q^{86} - q^{87} + ( -1 - 2 \beta_{1} + \beta_{3} + 2 \beta_{4} ) q^{88} + ( 1 - 4 \beta_{1} + 2 \beta_{3} + \beta_{4} ) q^{89} + ( 3 - 2 \beta_{1} - \beta_{4} ) q^{90} + ( 2 + \beta_{1} - 2 \beta_{3} ) q^{91} + ( 2 - \beta_{1} ) q^{92} + ( 2 + \beta_{2} - 2 \beta_{3} ) q^{93} + ( -7 + 6 \beta_{1} - 8 \beta_{4} ) q^{94} + ( -1 - 5 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} + 4 \beta_{4} ) q^{95} + ( 2 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} ) q^{96} + ( 1 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} ) q^{97} + ( 10 - 2 \beta_{1} + 3 \beta_{2} - \beta_{3} - 3 \beta_{4} ) q^{98} + ( -2 + \beta_{4} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5q - 2q^{2} - 5q^{3} + 8q^{4} - 3q^{5} + 2q^{6} - 5q^{7} - 3q^{8} + 5q^{9} + O(q^{10})$$ $$5q - 2q^{2} - 5q^{3} + 8q^{4} - 3q^{5} + 2q^{6} - 5q^{7} - 3q^{8} + 5q^{9} + 9q^{10} - 8q^{11} - 8q^{12} + 5q^{13} - 2q^{14} + 3q^{15} - 10q^{16} + 2q^{17} - 2q^{18} - 9q^{19} - 12q^{20} + 5q^{21} + 10q^{22} + 5q^{23} + 3q^{24} + 2q^{25} - 3q^{26} - 5q^{27} - 14q^{28} + 5q^{29} - 9q^{30} - 6q^{31} - 8q^{32} + 8q^{33} + 3q^{34} - 15q^{35} + 8q^{36} + 10q^{37} - 10q^{38} - 5q^{39} + 26q^{40} - 11q^{41} + 2q^{42} - 9q^{43} - 16q^{44} - 3q^{45} - 2q^{46} - 13q^{47} + 10q^{48} - 6q^{49} - 18q^{50} - 2q^{51} + 12q^{52} + q^{53} + 2q^{54} + 13q^{55} - 4q^{56} + 9q^{57} - 2q^{58} + 6q^{59} + 12q^{60} + 23q^{61} - 36q^{62} - 5q^{63} - q^{64} - 20q^{65} - 10q^{66} - 10q^{67} - 10q^{68} - 5q^{69} - 16q^{70} - 11q^{71} - 3q^{72} + 31q^{73} - 18q^{74} - 2q^{75} - 8q^{76} + 3q^{77} + 3q^{78} + 8q^{79} - 8q^{80} + 5q^{81} + 16q^{82} + 7q^{83} + 14q^{84} + 6q^{85} - 36q^{86} - 5q^{87} - 3q^{88} + 3q^{89} + 9q^{90} + 8q^{91} + 8q^{92} + 6q^{93} - 39q^{94} - 11q^{95} + 8q^{96} + 3q^{97} + 38q^{98} - 8q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - 2 x^{4} - 5 x^{3} + 11 x^{2} - x - 3$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{3} - 5 \nu + 1$$ $$\beta_{3}$$ $$=$$ $$\nu^{4} - \nu^{3} - 5 \nu^{2} + 5 \nu + 1$$ $$\beta_{4}$$ $$=$$ $$-\nu^{4} + \nu^{3} + 6 \nu^{2} - 5 \nu - 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{4} + \beta_{3} + 2$$ $$\nu^{3}$$ $$=$$ $$\beta_{2} + 5 \beta_{1} - 1$$ $$\nu^{4}$$ $$=$$ $$5 \beta_{4} + 6 \beta_{3} + \beta_{2} + 8$$

## 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
 −2.28064 0.762877 2.26093 1.70431 −0.447481
−2.50612 −1.00000 4.28064 −2.73973 2.50612 −1.54091 −5.71555 1.00000 6.86609
1.2 −1.79920 −1.00000 1.23712 −2.60753 1.79920 1.37040 1.37257 1.00000 4.69147
1.3 −1.31874 −1.00000 −0.260930 2.51371 1.31874 −2.25278 2.98157 1.00000 −3.31493
1.4 1.51515 −1.00000 0.295689 −1.86677 −1.51515 1.57109 −2.58229 1.00000 −2.82845
1.5 2.10891 −1.00000 2.44748 1.70032 −2.10891 −4.14780 0.943696 1.00000 3.58582
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$
$$23$$ $$-1$$
$$29$$ $$-1$$

## 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 2001.2.a.h 5
3.b odd 2 1 6003.2.a.h 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2001.2.a.h 5 1.a even 1 1 trivial
6003.2.a.h 5 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2001))$$:

 $$T_{2}^{5} + 2 T_{2}^{4} - 7 T_{2}^{3} - 13 T_{2}^{2} + 11 T_{2} + 19$$ $$T_{5}^{5} + 3 T_{5}^{4} - 9 T_{5}^{3} - 28 T_{5}^{2} + 17 T_{5} + 57$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$19 + 11 T - 13 T^{2} - 7 T^{3} + 2 T^{4} + T^{5}$$
$3$ $$( 1 + T )^{5}$$
$5$ $$57 + 17 T - 28 T^{2} - 9 T^{3} + 3 T^{4} + T^{5}$$
$7$ $$31 - T - 25 T^{2} - 2 T^{3} + 5 T^{4} + T^{5}$$
$11$ $$-1 - 4 T + 7 T^{2} + 16 T^{3} + 8 T^{4} + T^{5}$$
$13$ $$9 + 29 T + 23 T^{2} - 4 T^{3} - 5 T^{4} + T^{5}$$
$17$ $$-3 - T + 11 T^{2} - 5 T^{3} - 2 T^{4} + T^{5}$$
$19$ $$-563 - 699 T - 246 T^{2} - 6 T^{3} + 9 T^{4} + T^{5}$$
$23$ $$( -1 + T )^{5}$$
$29$ $$( -1 + T )^{5}$$
$31$ $$139 + 206 T - 59 T^{2} - 40 T^{3} + 6 T^{4} + T^{5}$$
$37$ $$-821 + 204 T + 203 T^{2} - 24 T^{3} - 10 T^{4} + T^{5}$$
$41$ $$-921 - 1307 T - 464 T^{2} - 21 T^{3} + 11 T^{4} + T^{5}$$
$43$ $$7447 + 1281 T - 626 T^{2} - 75 T^{3} + 9 T^{4} + T^{5}$$
$47$ $$49139 + 3908 T - 1877 T^{2} - 145 T^{3} + 13 T^{4} + T^{5}$$
$53$ $$361 + 551 T + 20 T^{2} - 49 T^{3} - T^{4} + T^{5}$$
$59$ $$1824 - 2896 T + 1408 T^{2} - 204 T^{3} - 6 T^{4} + T^{5}$$
$61$ $$-281 - 4217 T + 863 T^{2} + 84 T^{3} - 23 T^{4} + T^{5}$$
$67$ $$-513 - 1733 T - 1483 T^{2} - 133 T^{3} + 10 T^{4} + T^{5}$$
$71$ $$57 + 85 T - 119 T^{2} + 2 T^{3} + 11 T^{4} + T^{5}$$
$73$ $$1317 - 344 T - 669 T^{2} + 283 T^{3} - 31 T^{4} + T^{5}$$
$79$ $$-2921 + 332 T + 459 T^{2} - 62 T^{3} - 8 T^{4} + T^{5}$$
$83$ $$40743 + 17965 T + 982 T^{2} - 285 T^{3} - 7 T^{4} + T^{5}$$
$89$ $$-687 - 1279 T - 757 T^{2} - 150 T^{3} - 3 T^{4} + T^{5}$$
$97$ $$4337 - 5355 T + 2113 T^{2} - 268 T^{3} - 3 T^{4} + T^{5}$$