# Properties

 Label 2169.2.a.e Level $2169$ Weight $2$ Character orbit 2169.a Self dual yes Analytic conductor $17.320$ Analytic rank $0$ Dimension $7$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2169 = 3^{2} \cdot 241$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2169.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$17.3195521984$$ Analytic rank: $$0$$ Dimension: $$7$$ Coefficient field: 7.7.31056073.1 Defining polynomial: $$x^{7} - 3 x^{6} - 3 x^{5} + 11 x^{4} + x^{3} - 9 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 241) 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.

 $$f(q)$$ $$=$$ $$q + ( 1 - \beta_{1} ) q^{2} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{4} + ( 1 + \beta_{1} + \beta_{2} + \beta_{6} ) q^{5} + ( -2 + \beta_{2} + \beta_{3} + \beta_{5} - \beta_{6} ) q^{7} + ( 2 + \beta_{2} - 2 \beta_{3} + \beta_{5} + \beta_{6} ) q^{8} +O(q^{10})$$ $$q + ( 1 - \beta_{1} ) q^{2} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{4} + ( 1 + \beta_{1} + \beta_{2} + \beta_{6} ) q^{5} + ( -2 + \beta_{2} + \beta_{3} + \beta_{5} - \beta_{6} ) q^{7} + ( 2 + \beta_{2} - 2 \beta_{3} + \beta_{5} + \beta_{6} ) q^{8} + ( 1 - \beta_{1} - \beta_{4} + \beta_{6} ) q^{10} + ( 3 + \beta_{1} + \beta_{2} + 2 \beta_{4} + \beta_{5} + \beta_{6} ) q^{11} + ( 1 + \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{6} ) q^{13} + ( -1 + \beta_{3} - 2 \beta_{4} - \beta_{5} - 2 \beta_{6} ) q^{14} + ( 3 - 2 \beta_{3} + \beta_{4} + 2 \beta_{5} + 3 \beta_{6} ) q^{16} + ( -1 + \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} + 3 \beta_{5} - \beta_{6} ) q^{17} + ( -3 + 2 \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} ) q^{19} + ( 2 - 2 \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{6} ) q^{20} + ( 4 - 3 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{6} ) q^{22} + ( 2 - \beta_{1} + \beta_{3} - \beta_{4} - 2 \beta_{5} - 2 \beta_{6} ) q^{23} + ( 1 + \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{6} ) q^{25} + ( 3 - 2 \beta_{1} - \beta_{2} - 3 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + 3 \beta_{6} ) q^{26} + ( -\beta_{1} - 3 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} - \beta_{6} ) q^{28} + ( 2 - 3 \beta_{1} - \beta_{2} - 3 \beta_{4} - 4 \beta_{5} - 2 \beta_{6} ) q^{29} + ( -3 + 2 \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + \beta_{6} ) q^{31} + ( 4 + 2 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} + 3 \beta_{6} ) q^{32} + ( 1 - 2 \beta_{1} - 2 \beta_{4} - \beta_{5} - 2 \beta_{6} ) q^{34} + ( -3 - 2 \beta_{1} - \beta_{2} + \beta_{3} - 3 \beta_{4} + \beta_{5} - 4 \beta_{6} ) q^{35} + ( -2 + \beta_{1} - 3 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} - 2 \beta_{6} ) q^{37} + ( -7 + 2 \beta_{1} - 2 \beta_{2} + 5 \beta_{3} - 4 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} ) q^{38} + ( 4 - 2 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - \beta_{6} ) q^{40} + ( 2 - 3 \beta_{1} + 3 \beta_{3} + \beta_{4} - 3 \beta_{5} - \beta_{6} ) q^{41} + ( 5 - 5 \beta_{1} - 4 \beta_{3} - 3 \beta_{5} - 2 \beta_{6} ) q^{43} + ( 4 - 4 \beta_{1} + \beta_{2} - 5 \beta_{3} - \beta_{4} - \beta_{5} ) q^{44} + ( -4 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} - \beta_{5} - 3 \beta_{6} ) q^{46} + ( 2 + 5 \beta_{1} + 2 \beta_{2} - \beta_{3} + 3 \beta_{4} + \beta_{5} + 4 \beta_{6} ) q^{47} + ( -\beta_{1} - 4 \beta_{2} + \beta_{3} - 5 \beta_{4} - 4 \beta_{5} ) q^{49} + ( -2 \beta_{2} + 2 \beta_{3} - \beta_{5} ) q^{50} + ( 9 - \beta_{1} + \beta_{2} - 6 \beta_{3} + 2 \beta_{4} + \beta_{5} + 4 \beta_{6} ) q^{52} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} + 5 \beta_{4} + 4 \beta_{5} + \beta_{6} ) q^{53} + ( 5 + 5 \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} + 6 \beta_{6} ) q^{55} + ( -3 + 2 \beta_{1} + 3 \beta_{3} + 2 \beta_{4} + \beta_{6} ) q^{56} + ( 1 - 3 \beta_{1} + 3 \beta_{3} - 2 \beta_{4} - 2 \beta_{6} ) q^{58} + ( 4 - 3 \beta_{2} - \beta_{3} + 2 \beta_{4} + 4 \beta_{5} ) q^{59} + ( 2 + 3 \beta_{1} + 4 \beta_{2} - \beta_{3} + 7 \beta_{4} + 5 \beta_{5} + 2 \beta_{6} ) q^{61} + ( -3 + 3 \beta_{1} - \beta_{2} - \beta_{5} ) q^{62} + ( 3 - 3 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{64} + ( 3 + 2 \beta_{1} + 2 \beta_{2} - 5 \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{6} ) q^{65} + ( 6 + \beta_{2} + 5 \beta_{4} + 4 \beta_{5} + 5 \beta_{6} ) q^{67} + ( 4 - 5 \beta_{1} - 3 \beta_{2} - \beta_{3} - 4 \beta_{4} - 6 \beta_{5} ) q^{68} + ( -3 + 2 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} - \beta_{5} - 5 \beta_{6} ) q^{70} + ( 8 - 4 \beta_{1} - 2 \beta_{2} - \beta_{3} - 4 \beta_{4} - 3 \beta_{5} - \beta_{6} ) q^{71} + ( -1 + \beta_{1} + 4 \beta_{2} - 2 \beta_{3} - 3 \beta_{4} + 4 \beta_{5} ) q^{73} + ( -9 + 3 \beta_{1} - \beta_{2} + 7 \beta_{3} - 3 \beta_{4} - 4 \beta_{5} - 6 \beta_{6} ) q^{74} + ( -11 + 3 \beta_{1} - 4 \beta_{2} + 7 \beta_{3} - 5 \beta_{4} - 3 \beta_{5} - 7 \beta_{6} ) q^{76} + ( -4 - 2 \beta_{2} - 2 \beta_{4} + 3 \beta_{5} - 3 \beta_{6} ) q^{77} + ( -5 \beta_{1} - 3 \beta_{2} + \beta_{3} - 4 \beta_{5} ) q^{79} + ( 7 - 3 \beta_{1} - \beta_{2} - 2 \beta_{3} + 5 \beta_{4} + 3 \beta_{5} + 4 \beta_{6} ) q^{80} + ( 4 - 3 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} - 3 \beta_{5} - 4 \beta_{6} ) q^{82} + ( 3 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{6} ) q^{83} + ( -3 + \beta_{2} - 3 \beta_{4} + 4 \beta_{5} - 4 \beta_{6} ) q^{85} + ( 10 - 7 \beta_{1} - 4 \beta_{3} + 4 \beta_{4} + 4 \beta_{5} + 2 \beta_{6} ) q^{86} + ( 4 + \beta_{1} + 6 \beta_{2} - 6 \beta_{3} + \beta_{4} + 5 \beta_{5} + 3 \beta_{6} ) q^{88} + ( -4 + 9 \beta_{1} + 3 \beta_{2} + 4 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - \beta_{6} ) q^{89} + ( 1 - 7 \beta_{2} - \beta_{3} - \beta_{5} + 3 \beta_{6} ) q^{91} + ( -2 + \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 3 \beta_{6} ) q^{92} + ( -1 - 3 \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{5} + 5 \beta_{6} ) q^{94} + ( 3 - \beta_{1} - 3 \beta_{2} + 5 \beta_{3} - \beta_{4} - 5 \beta_{5} - 3 \beta_{6} ) q^{95} + ( 2 - 4 \beta_{2} + 3 \beta_{4} - 2 \beta_{5} + \beta_{6} ) q^{97} + ( -6 + 4 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{6} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$7q + 4q^{2} + 2q^{4} + 8q^{5} - 7q^{7} + 6q^{8} + O(q^{10})$$ $$7q + 4q^{2} + 2q^{4} + 8q^{5} - 7q^{7} + 6q^{8} + 3q^{10} + 18q^{11} - q^{13} + 6q^{14} + 4q^{16} + 2q^{17} - 6q^{19} + 8q^{20} + 10q^{22} + 22q^{23} + 5q^{25} - 8q^{26} + 9q^{28} + 16q^{29} - 18q^{31} + 6q^{32} + 11q^{34} - 7q^{35} + 8q^{37} - 16q^{38} + 14q^{40} + 15q^{41} + 14q^{43} + 4q^{44} + 11q^{46} + 10q^{47} + 6q^{49} + 4q^{50} + 27q^{52} - 15q^{53} + 29q^{55} - 13q^{56} + 17q^{58} + 18q^{59} + 4q^{61} - 13q^{62} + 2q^{64} + 7q^{65} + 18q^{67} + 15q^{68} + 8q^{70} + 50q^{71} - 10q^{74} - 20q^{76} - 17q^{77} - 15q^{79} + 11q^{80} + 45q^{82} + 24q^{83} - 2q^{85} + 23q^{86} + 8q^{88} + 13q^{89} - 12q^{91} + 10q^{92} - 32q^{94} + 41q^{95} + q^{97} - 9q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{7} - 3 x^{6} - 3 x^{5} + 11 x^{4} + x^{3} - 9 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{5} - 2 \nu^{4} - 4 \nu^{3} + 5 \nu^{2} + 2 \nu - 1$$ $$\beta_{3}$$ $$=$$ $$\nu^{5} - 2 \nu^{4} - 4 \nu^{3} + 4 \nu^{2} + 3 \nu + 1$$ $$\beta_{4}$$ $$=$$ $$\nu^{6} - 3 \nu^{5} - 2 \nu^{4} + 9 \nu^{3} - 3 \nu^{2} - 4 \nu + 2$$ $$\beta_{5}$$ $$=$$ $$\nu^{6} - 2 \nu^{5} - 5 \nu^{4} + 6 \nu^{3} + 7 \nu^{2} - 3 \nu - 2$$ $$\beta_{6}$$ $$=$$ $$-\nu^{6} + 3 \nu^{5} + 3 \nu^{4} - 11 \nu^{3} - \nu^{2} + 8 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{3} + \beta_{2} + \beta_{1} + 2$$ $$\nu^{3}$$ $$=$$ $$-\beta_{6} - \beta_{5} - \beta_{3} + 2 \beta_{2} + 4 \beta_{1} + 1$$ $$\nu^{4}$$ $$=$$ $$-\beta_{6} - 2 \beta_{5} + \beta_{4} - 6 \beta_{3} + 8 \beta_{2} + 8 \beta_{1} + 8$$ $$\nu^{5}$$ $$=$$ $$-6 \beta_{6} - 8 \beta_{5} + 2 \beta_{4} - 11 \beta_{3} + 20 \beta_{2} + 25 \beta_{1} + 11$$ $$\nu^{6}$$ $$=$$ $$-11 \beta_{6} - 19 \beta_{5} + 9 \beta_{4} - 39 \beta_{3} + 61 \beta_{2} + 62 \beta_{1} + 44$$

## 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
 2.73684 1.48734 1.27758 0.369356 −0.356270 −0.911223 −1.60363
−1.73684 0 1.01662 2.63180 0 −2.01025 1.70797 0 −4.57103
1.2 −0.487343 0 −1.76250 −0.961999 0 −4.61392 1.83363 0 0.468824
1.3 −0.277577 0 −1.92295 1.23324 0 1.36627 1.08892 0 −0.342320
1.4 0.630644 0 −1.60229 3.89634 0 −3.68231 −2.27176 0 2.45721
1.5 1.35627 0 −0.160532 −2.74184 0 −0.283608 −2.93026 0 −3.71867
1.6 1.91122 0 1.65278 2.25110 0 3.52970 −0.663624 0 4.30235
1.7 2.60363 0 4.77887 1.69135 0 −1.30586 7.23513 0 4.40364
 $$n$$: 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
$$3$$ $$-1$$
$$241$$ $$1$$

## 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 2169.2.a.e 7
3.b odd 2 1 241.2.a.a 7
12.b even 2 1 3856.2.a.j 7
15.d odd 2 1 6025.2.a.f 7

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
241.2.a.a 7 3.b odd 2 1
2169.2.a.e 7 1.a even 1 1 trivial
3856.2.a.j 7 12.b even 2 1
6025.2.a.f 7 15.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{7} - 4 T_{2}^{6} + 14 T_{2}^{4} - 10 T_{2}^{3} - 6 T_{2}^{2} + 3 T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2169))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 3 T - 6 T^{2} - 10 T^{3} + 14 T^{4} - 4 T^{6} + T^{7}$$
$3$ $$T^{7}$$
$5$ $$-127 + 137 T + 93 T^{2} - 165 T^{3} + 50 T^{4} + 12 T^{5} - 8 T^{6} + T^{7}$$
$7$ $$61 + 260 T + 127 T^{2} - 138 T^{3} - 98 T^{4} - 3 T^{5} + 7 T^{6} + T^{7}$$
$11$ $$-1069 - 1281 T + 1559 T^{2} - 137 T^{3} - 283 T^{4} + 117 T^{5} - 18 T^{6} + T^{7}$$
$13$ $$-1 + 13 T + 860 T^{2} + 533 T^{3} - 62 T^{4} - 48 T^{5} + T^{6} + T^{7}$$
$17$ $$1039 - 633 T - 1382 T^{2} + 967 T^{3} + 86 T^{4} - 65 T^{5} - 2 T^{6} + T^{7}$$
$19$ $$-5983 - 6849 T + 3337 T^{2} + 1067 T^{3} - 276 T^{4} - 56 T^{5} + 6 T^{6} + T^{7}$$
$23$ $$-1369 - 532 T + 1693 T^{2} - 75 T^{3} - 463 T^{4} + 168 T^{5} - 22 T^{6} + T^{7}$$
$29$ $$10769 - 23012 T + 17808 T^{2} - 6173 T^{3} + 839 T^{4} + 25 T^{5} - 16 T^{6} + T^{7}$$
$31$ $$-617 - 3770 T - 3600 T^{2} - 1006 T^{3} + 109 T^{4} + 104 T^{5} + 18 T^{6} + T^{7}$$
$37$ $$-78167 + 88791 T - 27243 T^{2} - 828 T^{3} + 1330 T^{4} - 119 T^{5} - 8 T^{6} + T^{7}$$
$41$ $$-101009 - 157642 T - 58348 T^{2} + 5058 T^{3} + 1974 T^{4} - 122 T^{5} - 15 T^{6} + T^{7}$$
$43$ $$296569 + 67463 T - 74048 T^{2} - 49 T^{3} + 2460 T^{4} - 141 T^{5} - 14 T^{6} + T^{7}$$
$47$ $$7793 - 48600 T - 18600 T^{2} + 3904 T^{3} + 997 T^{4} - 116 T^{5} - 10 T^{6} + T^{7}$$
$53$ $$230663 + 281202 T + 84407 T^{2} - 35 T^{3} - 2311 T^{4} - 123 T^{5} + 15 T^{6} + T^{7}$$
$59$ $$2076763 - 114293 T - 152610 T^{2} + 5036 T^{3} + 3030 T^{4} - 130 T^{5} - 18 T^{6} + T^{7}$$
$61$ $$23149 + 291498 T - 136478 T^{2} + 13144 T^{3} + 1693 T^{4} - 254 T^{5} - 4 T^{6} + T^{7}$$
$67$ $$2288147 + 4468 T - 189536 T^{2} + 4503 T^{3} + 3579 T^{4} - 157 T^{5} - 18 T^{6} + T^{7}$$
$71$ $$255937 - 122885 T - 37139 T^{2} + 34990 T^{3} - 8586 T^{4} + 955 T^{5} - 50 T^{6} + T^{7}$$
$73$ $$11879 - 5297 T - 192681 T^{2} + 37009 T^{3} + 1068 T^{4} - 378 T^{5} + T^{7}$$
$79$ $$52709 + 79692 T + 33855 T^{2} + 711 T^{3} - 1557 T^{4} - 85 T^{5} + 15 T^{6} + T^{7}$$
$83$ $$-4333 + 2523 T + 1594 T^{2} - 1238 T^{3} + 14 T^{4} + 124 T^{5} - 24 T^{6} + T^{7}$$
$89$ $$89477 + 26169 T - 119043 T^{2} + 12156 T^{3} + 4667 T^{4} - 363 T^{5} - 13 T^{6} + T^{7}$$
$97$ $$40121 - 136236 T - 43003 T^{2} + 15061 T^{3} + 939 T^{4} - 283 T^{5} - T^{6} + T^{7}$$