# Properties

 Label 1859.2.a.k Level $1859$ Weight $2$ Character orbit 1859.a Self dual yes Analytic conductor $14.844$ Analytic rank $1$ Dimension $6$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1859 = 11 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1859.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$14.8441897358$$ Analytic rank: $$1$$ Dimension: $$6$$ Coefficient field: 6.6.28561300.1 Defining polynomial: $$x^{6} - 9 x^{4} - x^{3} + 22 x^{2} + 4 x - 12$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 143) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 9 x^{4} - x^{3} + 22 x^{2} + 4 x - 12$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - 4 \nu$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{5} - 7 \nu^{3} - \nu^{2} + 10 \nu + 2$$$$)/2$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{5} + 2 \nu^{4} + 7 \nu^{3} - 13 \nu^{2} - 10 \nu + 16$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + 4 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{5} + \beta_{4} + 7 \beta_{2} + 12$$ $$\nu^{5}$$ $$=$$ $$2 \beta_{4} + 7 \beta_{3} + \beta_{2} + 18 \beta_{1} + 1$$

## 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
 0.736891 −1.08549 −1.58147 1.80787 −2.23039 2.35258
−2.45699 −0.664116 4.03681 −1.73689 1.63173 −1.63580 −5.00442 −2.55895 4.26753
1.2 −1.82172 3.11527 1.31865 0.0854874 −5.67515 −4.51942 1.24122 6.70494 −0.155734
1.3 −0.498953 −0.240597 −1.75105 0.581470 0.120047 1.41017 1.87160 −2.94211 −0.290126
1.4 0.268396 2.35117 −1.92796 −2.80787 0.631044 2.38467 −1.05425 2.52800 −0.753621
1.5 1.97462 −0.571492 1.89914 1.23039 −1.12848 −2.55803 −0.199164 −2.67340 2.42955
1.6 2.53464 −2.99024 4.42441 −3.35258 −7.57919 1.91841 6.14501 5.94153 −8.49760
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$11$$ $$1$$
$$13$$ $$1$$

## 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 1859.2.a.k 6
13.b even 2 1 1859.2.a.l 6
13.c even 3 2 143.2.e.c 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
143.2.e.c 12 13.c even 3 2
1859.2.a.k 6 1.a even 1 1 trivial
1859.2.a.l 6 13.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1859))$$:

 $$T_{2}^{6} - 10 T_{2}^{4} - T_{2}^{3} + 24 T_{2}^{2} + 5 T_{2} - 3$$ $$T_{7}^{6} + 3 T_{7}^{5} - 16 T_{7}^{4} - 27 T_{7}^{3} + 82 T_{7}^{2} + 52 T_{7} - 122$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-3 + 5 T + 24 T^{2} - T^{3} - 10 T^{4} + T^{6}$$
$3$ $$2 + 14 T + 26 T^{2} + 7 T^{3} - 12 T^{4} - T^{5} + T^{6}$$
$5$ $$-1 + 13 T - 14 T^{2} - 15 T^{3} + 6 T^{4} + 6 T^{5} + T^{6}$$
$7$ $$-122 + 52 T + 82 T^{2} - 27 T^{3} - 16 T^{4} + 3 T^{5} + T^{6}$$
$11$ $$( 1 + T )^{6}$$
$13$ $$T^{6}$$
$17$ $$-139 + 191 T + 260 T^{2} + 13 T^{3} - 34 T^{4} - 2 T^{5} + T^{6}$$
$19$ $$-302 + 1034 T + 42 T^{2} - 211 T^{3} - 9 T^{4} + 10 T^{5} + T^{6}$$
$23$ $$592 + 1736 T + 904 T^{2} + 15 T^{3} - 56 T^{4} - 3 T^{5} + T^{6}$$
$29$ $$-19891 - 528 T + 3700 T^{2} + 197 T^{3} - 126 T^{4} - 3 T^{5} + T^{6}$$
$31$ $$120 - 380 T + 350 T^{2} - 57 T^{3} - 31 T^{4} + 5 T^{5} + T^{6}$$
$37$ $$-5171 - 9344 T - 4290 T^{2} - 81 T^{3} + 176 T^{4} + 25 T^{5} + T^{6}$$
$41$ $$-6427 - 6715 T - 1642 T^{2} + 315 T^{3} + 182 T^{4} + 24 T^{5} + T^{6}$$
$43$ $$1448 + 2268 T + 546 T^{2} - 309 T^{3} - 43 T^{4} + 8 T^{5} + T^{6}$$
$47$ $$-2 + 164 T^{2} - 183 T^{3} - 15 T^{4} + 10 T^{5} + T^{6}$$
$53$ $$121 + 1177 T - 4650 T^{2} + 1997 T^{3} - 156 T^{4} - 10 T^{5} + T^{6}$$
$59$ $$-18398 - 12496 T + 5792 T^{2} + 469 T^{3} - 149 T^{4} - 4 T^{5} + T^{6}$$
$61$ $$-655 - 3250 T - 1010 T^{2} + 243 T^{3} + 144 T^{4} + 21 T^{5} + T^{6}$$
$67$ $$2978 + 3266 T - 2142 T^{2} - 391 T^{3} + 94 T^{4} + 21 T^{5} + T^{6}$$
$71$ $$-101256 - 25820 T + 15530 T^{2} + 911 T^{3} - 272 T^{4} - 3 T^{5} + T^{6}$$
$73$ $$296 - 196 T - 254 T^{2} + 5 T^{3} + 49 T^{4} + 13 T^{5} + T^{6}$$
$79$ $$-1572016 + 36648 T + 48776 T^{2} - 1034 T^{3} - 408 T^{4} + 4 T^{5} + T^{6}$$
$83$ $$-12114 - 11794 T + 7360 T^{2} - 397 T^{3} - 177 T^{4} + 8 T^{5} + T^{6}$$
$89$ $$648364 - 288808 T + 17206 T^{2} + 3699 T^{3} - 328 T^{4} - 9 T^{5} + T^{6}$$
$97$ $$-7628 + 15440 T + 9290 T^{2} - 3091 T^{3} - 267 T^{4} + 15 T^{5} + T^{6}$$