# Properties

 Label 2366.2.a.be Level 2366 Weight 2 Character orbit 2366.a Self dual yes Analytic conductor 18.893 Analytic rank 1 Dimension 6 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2366 = 2 \cdot 7 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2366.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$18.8926051182$$ Analytic rank: $$1$$ Dimension: $$6$$ Coefficient field: 6.6.6052921.1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} - 11 x^{4} + 7 x^{3} + 33 x^{2} - 9 x - 27$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{4} - \nu^{3} - 8 \nu^{2} + 4 \nu + 12$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{5} - 4 \nu^{4} - 8 \nu^{3} + 31 \nu^{2} + 12 \nu - 36$$$$)/9$$ $$\beta_{4}$$ $$=$$ $$($$$$4 \nu^{5} - 4 \nu^{4} - 35 \nu^{3} + 19 \nu^{2} + 60 \nu$$$$)/9$$ $$\beta_{5}$$ $$=$$ $$($$$$4 \nu^{5} - 7 \nu^{4} - 32 \nu^{3} + 52 \nu^{2} + 39 \nu - 72$$$$)/9$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{5} - \beta_{4} + \beta_{2} + \beta_{1} + 4$$ $$\nu^{3}$$ $$=$$ $$\beta_{5} - 4 \beta_{3} - 3 \beta_{2} + 5 \beta_{1} + 4$$ $$\nu^{4}$$ $$=$$ $$9 \beta_{5} - 8 \beta_{4} - 4 \beta_{3} + 8 \beta_{2} + 9 \beta_{1} + 24$$ $$\nu^{5}$$ $$=$$ $$13 \beta_{5} - \beta_{4} - 39 \beta_{3} - 23 \beta_{2} + 33 \beta_{1} + 40$$

## 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
 −2.46434 −1.52377 −1.05140 1.21736 1.96881 2.85334
−1.00000 −2.46434 1.00000 −3.19361 2.46434 −1.00000 −1.00000 3.07299 3.19361
1.2 −1.00000 −1.52377 1.00000 1.45506 1.52377 −1.00000 −1.00000 −0.678139 −1.45506
1.3 −1.00000 −1.05140 1.00000 −2.26985 1.05140 −1.00000 −1.00000 −1.89456 2.26985
1.4 −1.00000 1.21736 1.00000 3.44059 −1.21736 −1.00000 −1.00000 −1.51803 −3.44059
1.5 −1.00000 1.96881 1.00000 −2.90010 −1.96881 −1.00000 −1.00000 0.876202 2.90010
1.6 −1.00000 2.85334 1.00000 −0.532083 −2.85334 −1.00000 −1.00000 5.14154 0.532083
 $$n$$: 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

## 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 2366.2.a.be 6
13.b even 2 1 2366.2.a.bg yes 6
13.d odd 4 2 2366.2.d.q 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2366.2.a.be 6 1.a even 1 1 trivial
2366.2.a.bg yes 6 13.b even 2 1
2366.2.d.q 12 13.d odd 4 2

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$7$$ $$1$$
$$13$$ $$1$$

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2366))$$:

 $$T_{3}^{6} - T_{3}^{5} - 11 T_{3}^{4} + 7 T_{3}^{3} + 33 T_{3}^{2} - 9 T_{3} - 27$$ $$T_{5}^{6} + 4 T_{5}^{5} - 11 T_{5}^{4} - 57 T_{5}^{3} - 14 T_{5}^{2} + 112 T_{5} + 56$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{6}$$
$3$ $$1 - T + 7 T^{2} - 8 T^{3} + 36 T^{4} - 36 T^{5} + 117 T^{6} - 108 T^{7} + 324 T^{8} - 216 T^{9} + 567 T^{10} - 243 T^{11} + 729 T^{12}$$
$5$ $$1 + 4 T + 19 T^{2} + 43 T^{3} + 141 T^{4} + 257 T^{5} + 766 T^{6} + 1285 T^{7} + 3525 T^{8} + 5375 T^{9} + 11875 T^{10} + 12500 T^{11} + 15625 T^{12}$$
$7$ $$( 1 + T )^{6}$$
$11$ $$1 + 6 T + 61 T^{2} + 270 T^{3} + 1565 T^{4} + 5398 T^{5} + 22427 T^{6} + 59378 T^{7} + 189365 T^{8} + 359370 T^{9} + 893101 T^{10} + 966306 T^{11} + 1771561 T^{12}$$
$13$ 1
$17$ $$1 + 9 T + 101 T^{2} + 638 T^{3} + 4134 T^{4} + 19764 T^{5} + 92011 T^{6} + 335988 T^{7} + 1194726 T^{8} + 3134494 T^{9} + 8435621 T^{10} + 12778713 T^{11} + 24137569 T^{12}$$
$19$ $$1 + 10 T + 97 T^{2} + 568 T^{3} + 3695 T^{4} + 17488 T^{5} + 89105 T^{6} + 332272 T^{7} + 1333895 T^{8} + 3895912 T^{9} + 12641137 T^{10} + 24760990 T^{11} + 47045881 T^{12}$$
$23$ $$1 - 21 T + 244 T^{2} - 1974 T^{3} + 13037 T^{4} - 74557 T^{5} + 380660 T^{6} - 1714811 T^{7} + 6896573 T^{8} - 24017658 T^{9} + 68281204 T^{10} - 135163203 T^{11} + 148035889 T^{12}$$
$29$ $$1 + T + 60 T^{2} - 38 T^{3} + 1813 T^{4} - 5663 T^{5} + 52956 T^{6} - 164227 T^{7} + 1524733 T^{8} - 926782 T^{9} + 42436860 T^{10} + 20511149 T^{11} + 594823321 T^{12}$$
$31$ $$1 + 20 T + 321 T^{2} + 3425 T^{3} + 31041 T^{4} + 220905 T^{5} + 1365586 T^{6} + 6848055 T^{7} + 29830401 T^{8} + 102034175 T^{9} + 296450241 T^{10} + 572583020 T^{11} + 887503681 T^{12}$$
$37$ $$1 + 16 T + 179 T^{2} + 1601 T^{3} + 12575 T^{4} + 87987 T^{5} + 557826 T^{6} + 3255519 T^{7} + 17215175 T^{8} + 81095453 T^{9} + 335474819 T^{10} + 1109503312 T^{11} + 2565726409 T^{12}$$
$41$ $$1 + 2 T + 77 T^{2} - 18 T^{3} + 4625 T^{4} + 618 T^{5} + 215007 T^{6} + 25338 T^{7} + 7774625 T^{8} - 1240578 T^{9} + 217583597 T^{10} + 231712402 T^{11} + 4750104241 T^{12}$$
$43$ $$1 - 2 T + 25 T^{2} - 136 T^{3} + 5603 T^{4} - 9604 T^{5} + 92961 T^{6} - 412972 T^{7} + 10359947 T^{8} - 10812952 T^{9} + 85470025 T^{10} - 294016886 T^{11} + 6321363049 T^{12}$$
$47$ $$1 - 5 T + 164 T^{2} - 954 T^{3} + 15413 T^{4} - 75645 T^{5} + 908724 T^{6} - 3555315 T^{7} + 34047317 T^{8} - 99047142 T^{9} + 800267684 T^{10} - 1146725035 T^{11} + 10779215329 T^{12}$$
$53$ $$1 - 28 T + 551 T^{2} - 7735 T^{3} + 88485 T^{4} - 826301 T^{5} + 6574358 T^{6} - 43793953 T^{7} + 248554365 T^{8} - 1151563595 T^{9} + 4347655031 T^{10} - 11709473804 T^{11} + 22164361129 T^{12}$$
$59$ $$1 - 12 T + 243 T^{2} - 1492 T^{3} + 18371 T^{4} - 52238 T^{5} + 891263 T^{6} - 3082042 T^{7} + 63949451 T^{8} - 306425468 T^{9} + 2944518723 T^{10} - 8579091588 T^{11} + 42180533641 T^{12}$$
$61$ $$1 + 27 T + 598 T^{2} + 8766 T^{3} + 111115 T^{4} + 1098335 T^{5} + 9564612 T^{6} + 66998435 T^{7} + 413458915 T^{8} + 1989715446 T^{9} + 8279812918 T^{10} + 22804100127 T^{11} + 51520374361 T^{12}$$
$67$ $$1 + 16 T + 259 T^{2} + 2960 T^{3} + 34347 T^{4} + 316274 T^{5} + 2882699 T^{6} + 21190358 T^{7} + 154183683 T^{8} + 890258480 T^{9} + 5219140339 T^{10} + 21602001712 T^{11} + 90458382169 T^{12}$$
$71$ $$1 + 169 T^{2} + 35 T^{3} + 11271 T^{4} - 4445 T^{5} + 581862 T^{6} - 315595 T^{7} + 56817111 T^{8} + 12526885 T^{9} + 4294574089 T^{10} + 128100283921 T^{12}$$
$73$ $$1 + 38 T + 811 T^{2} + 12294 T^{3} + 150495 T^{4} + 1568896 T^{5} + 14314189 T^{6} + 114529408 T^{7} + 801987855 T^{8} + 4782574998 T^{9} + 23030973451 T^{10} + 78776720534 T^{11} + 151334226289 T^{12}$$
$79$ $$1 - 6 T + 337 T^{2} - 1579 T^{3} + 54647 T^{4} - 210917 T^{5} + 5415446 T^{6} - 16662443 T^{7} + 341051927 T^{8} - 778508581 T^{9} + 13126177297 T^{10} - 18462338394 T^{11} + 243087455521 T^{12}$$
$83$ $$1 - 6 T + 265 T^{2} - 1224 T^{3} + 35891 T^{4} - 128808 T^{5} + 3418761 T^{6} - 10691064 T^{7} + 247253099 T^{8} - 699867288 T^{9} + 12576455065 T^{10} - 23634243858 T^{11} + 326940373369 T^{12}$$
$89$ $$1 - T + 229 T^{2} + 94 T^{3} + 31324 T^{4} - 21644 T^{5} + 3356869 T^{6} - 1926316 T^{7} + 248117404 T^{8} + 66267086 T^{9} + 14367973189 T^{10} - 5584059449 T^{11} + 496981290961 T^{12}$$
$97$ $$1 + 16 T + 511 T^{2} + 6198 T^{3} + 113769 T^{4} + 1087256 T^{5} + 14310613 T^{6} + 105463832 T^{7} + 1070452521 T^{8} + 5656747254 T^{9} + 45238462591 T^{10} + 137397444112 T^{11} + 832972004929 T^{12}$$