# Properties

 Label 2009.2.a.p Level $2009$ Weight $2$ Character orbit 2009.a Self dual yes Analytic conductor $16.042$ Analytic rank $1$ Dimension $7$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2009 = 7^{2} \cdot 41$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2009.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$16.0419457661$$ Analytic rank: $$1$$ Dimension: $$7$$ Coefficient field: $$\mathbb{Q}[x]/(x^{7} - \cdots)$$ Defining polynomial: $$x^{7} - 10 x^{5} - x^{4} + 30 x^{3} + 7 x^{2} - 25 x - 11$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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 $$1,\beta_1,\ldots,\beta_{6}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{7} - 10 x^{5} - x^{4} + 30 x^{3} + 7 x^{2} - 25 x - 11$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - 4 \nu$$ $$\beta_{4}$$ $$=$$ $$\nu^{4} - 5 \nu^{2} + 3$$ $$\beta_{5}$$ $$=$$ $$\nu^{5} - 7 \nu^{3} - \nu^{2} + 10 \nu + 4$$ $$\beta_{6}$$ $$=$$ $$\nu^{6} - \nu^{5} - 9 \nu^{4} + 7 \nu^{3} + 23 \nu^{2} - 11 \nu - 14$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + 4 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{4} + 5 \beta_{2} + 12$$ $$\nu^{5}$$ $$=$$ $$\beta_{5} + 7 \beta_{3} + \beta_{2} + 18 \beta_{1} - 1$$ $$\nu^{6}$$ $$=$$ $$\beta_{6} + \beta_{5} + 9 \beta_{4} + 23 \beta_{2} + \beta_{1} + 52$$

## 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
 −0.522645 −0.907606 1.28109 −1.84648 1.95500 −2.21905 2.25969
−2.72684 −1.52265 5.43567 2.93285 4.15201 0 −9.36852 −0.681552 −7.99743
1.2 −2.17625 −1.90761 2.73607 −2.13420 4.15143 0 −1.60187 0.638961 4.64455
1.3 −1.35880 0.281095 −0.153673 −3.77273 −0.381951 0 2.92640 −2.92099 5.12637
1.4 0.409485 −2.84648 −1.83232 2.40160 −1.16559 0 −1.56928 5.10244 0.983421
1.5 0.822027 0.955000 −1.32427 3.05691 0.785036 0 −2.73264 −2.08797 2.51286
1.6 1.92419 −3.21905 1.70251 −2.13009 −6.19407 0 −0.572424 7.36229 −4.09870
1.7 2.10619 1.25969 2.43602 −4.35435 2.65313 0 0.918336 −1.41319 −9.17108
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.7 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$7$$ $$-1$$
$$41$$ $$-1$$

## 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 2009.2.a.p 7
7.b odd 2 1 2009.2.a.q yes 7

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2009.2.a.p 7 1.a even 1 1 trivial
2009.2.a.q yes 7 7.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2009))$$:

 $$T_{2}^{7} + T_{2}^{6} - 11 T_{2}^{5} - 6 T_{2}^{4} + 37 T_{2}^{3} + 4 T_{2}^{2} - 34 T_{2} + 11$$ $$T_{3}^{7} + 7 T_{3}^{6} + 11 T_{3}^{5} - 16 T_{3}^{4} - 39 T_{3}^{3} + 12 T_{3}^{2} + 32 T_{3} - 9$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$11 - 34 T + 4 T^{2} + 37 T^{3} - 6 T^{4} - 11 T^{5} + T^{6} + T^{7}$$
$3$ $$-9 + 32 T + 12 T^{2} - 39 T^{3} - 16 T^{4} + 11 T^{5} + 7 T^{6} + T^{7}$$
$5$ $$-1608 - 560 T + 674 T^{2} + 209 T^{3} - 92 T^{4} - 25 T^{5} + 4 T^{6} + T^{7}$$
$7$ $$T^{7}$$
$11$ $$-152 - 364 T + 40 T^{2} + 281 T^{3} + 32 T^{4} - 42 T^{5} + T^{7}$$
$13$ $$-57 - 170 T + 343 T^{2} + 229 T^{3} - 47 T^{4} - 32 T^{5} + T^{6} + T^{7}$$
$17$ $$-1257 + 2510 T - 825 T^{2} - 559 T^{3} + 261 T^{4} + 4 T^{5} - 11 T^{6} + T^{7}$$
$19$ $$-18651 + 14000 T + 12652 T^{2} + 469 T^{3} - 668 T^{4} - 61 T^{5} + 9 T^{6} + T^{7}$$
$23$ $$17431 - 5030 T - 3148 T^{2} + 933 T^{3} + 178 T^{4} - 55 T^{5} - 3 T^{6} + T^{7}$$
$29$ $$-125048 + 40768 T + 22462 T^{2} - 1903 T^{3} - 1190 T^{4} - 41 T^{5} + 14 T^{6} + T^{7}$$
$31$ $$354168 - 4636 T - 56248 T^{2} - 8587 T^{3} + 1174 T^{4} + 391 T^{5} + 34 T^{6} + T^{7}$$
$37$ $$23664 + 17216 T - 28692 T^{2} + 4452 T^{3} + 903 T^{4} - 138 T^{5} - 7 T^{6} + T^{7}$$
$41$ $$( -1 + T )^{7}$$
$43$ $$-254433 - 243680 T + 29589 T^{2} + 12609 T^{3} - 663 T^{4} - 210 T^{5} + 3 T^{6} + T^{7}$$
$47$ $$-355008 + 17888 T + 41032 T^{2} - 284 T^{3} - 1483 T^{4} - 40 T^{5} + 17 T^{6} + T^{7}$$
$53$ $$152 - 2340 T + 2492 T^{2} - 319 T^{3} - 468 T^{4} + 187 T^{5} - 24 T^{6} + T^{7}$$
$59$ $$-25224 - 12568 T + 4150 T^{2} + 2161 T^{3} - 200 T^{4} - 93 T^{5} + 4 T^{6} + T^{7}$$
$61$ $$-216 - 3164 T - 9960 T^{2} - 5823 T^{3} - 1008 T^{4} + 10 T^{5} + 16 T^{6} + T^{7}$$
$67$ $$-467392 + 324912 T + 34624 T^{2} - 22631 T^{3} - 3608 T^{4} + 3 T^{5} + 24 T^{6} + T^{7}$$
$71$ $$13768 + 49556 T - 55256 T^{2} + 16869 T^{3} - 996 T^{4} - 206 T^{5} + 12 T^{6} + T^{7}$$
$73$ $$26136 - 27976 T - 3714 T^{2} + 7641 T^{3} - 1028 T^{4} - 130 T^{5} + 14 T^{6} + T^{7}$$
$79$ $$862592 - 546496 T - 9184 T^{2} + 23936 T^{3} - 776 T^{4} - 280 T^{5} + 8 T^{6} + T^{7}$$
$83$ $$7460904 + 3554048 T + 510094 T^{2} - 1423 T^{3} - 5732 T^{4} - 290 T^{5} + 14 T^{6} + T^{7}$$
$89$ $$157959 - 10378 T - 38998 T^{2} + 3019 T^{3} + 1922 T^{4} - 87 T^{5} - 19 T^{6} + T^{7}$$
$97$ $$-137463 + 158306 T + 4990 T^{2} - 17753 T^{3} - 2782 T^{4} + 17 T^{5} + 23 T^{6} + T^{7}$$