# Properties

 Label 2366.2.a.bf Level 2366 Weight 2 Character orbit 2366.a Self dual yes Analytic conductor 18.893 Analytic rank 0 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: $$0$$ Dimension: $$6$$ Coefficient field: 6.6.285686784.1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 182) 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2 x^{5} - 10 x^{4} + 12 x^{3} + 21 x^{2} + 2 x - 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{5} + 4 \nu^{4} + 7 \nu^{3} - 26 \nu^{2} - 9 \nu + 1$$$$)/5$$ $$\beta_{3}$$ $$=$$ $$($$$$-2 \nu^{5} + 3 \nu^{4} + 24 \nu^{3} - 17 \nu^{2} - 68 \nu - 13$$$$)/5$$ $$\beta_{4}$$ $$=$$ $$($$$$3 \nu^{5} - 7 \nu^{4} - 26 \nu^{3} + 43 \nu^{2} + 37 \nu - 3$$$$)/5$$ $$\beta_{5}$$ $$=$$ $$($$$$4 \nu^{5} - 11 \nu^{4} - 33 \nu^{3} + 74 \nu^{2} + 41 \nu - 24$$$$)/5$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{5} - \beta_{4} + \beta_{2} + \beta_{1} + 4$$ $$\nu^{3}$$ $$=$$ $$\beta_{4} + \beta_{3} + \beta_{2} + 8 \beta_{1} + 3$$ $$\nu^{4}$$ $$=$$ $$7 \beta_{5} - 5 \beta_{4} + \beta_{3} + 11 \beta_{2} + 13 \beta_{1} + 31$$ $$\nu^{5}$$ $$=$$ $$2 \beta_{5} + 13 \beta_{4} + 11 \beta_{3} + 20 \beta_{2} + 73 \beta_{1} + 42$$

## 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
 −2.55629 −0.865515 −0.466545 0.252878 2.29079 3.34469
−1.00000 −2.55629 1.00000 3.48754 2.55629 −1.00000 −1.00000 3.53463 −3.48754
1.2 −1.00000 −0.865515 1.00000 3.71131 0.865515 −1.00000 −1.00000 −2.25088 −3.71131
1.3 −1.00000 −0.466545 1.00000 −3.38938 0.466545 −1.00000 −1.00000 −2.78234 3.38938
1.4 −1.00000 0.252878 1.00000 −1.14776 −0.252878 −1.00000 −1.00000 −2.93605 1.14776
1.5 −1.00000 2.29079 1.00000 0.901839 −2.29079 −1.00000 −1.00000 2.24770 −0.901839
1.6 −1.00000 3.34469 1.00000 −1.56356 −3.34469 −1.00000 −1.00000 8.18694 1.56356
 $$n$$: 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.bf 6
13.b even 2 1 2366.2.a.bh 6
13.d odd 4 2 2366.2.d.r 12
13.f odd 12 2 182.2.m.b 12
39.k even 12 2 1638.2.bj.g 12
52.l even 12 2 1456.2.cc.d 12
91.w even 12 2 1274.2.v.d 12
91.x odd 12 2 1274.2.o.d 12
91.ba even 12 2 1274.2.o.e 12
91.bc even 12 2 1274.2.m.c 12
91.bd odd 12 2 1274.2.v.e 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
182.2.m.b 12 13.f odd 12 2
1274.2.m.c 12 91.bc even 12 2
1274.2.o.d 12 91.x odd 12 2
1274.2.o.e 12 91.ba even 12 2
1274.2.v.d 12 91.w even 12 2
1274.2.v.e 12 91.bd odd 12 2
1456.2.cc.d 12 52.l even 12 2
1638.2.bj.g 12 39.k even 12 2
2366.2.a.bf 6 1.a even 1 1 trivial
2366.2.a.bh 6 13.b even 2 1
2366.2.d.r 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2366))$$:

 $$T_{3}^{6} - 2 T_{3}^{5} - 10 T_{3}^{4} + 12 T_{3}^{3} + 21 T_{3}^{2} + 2 T_{3} - 2$$ $$T_{5}^{6} - 2 T_{5}^{5} - 19 T_{5}^{4} + 24 T_{5}^{3} + 93 T_{5}^{2} - 10 T_{5} - 71$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{6}$$
$3$ $$1 - 2 T + 8 T^{2} - 18 T^{3} + 36 T^{4} - 70 T^{5} + 124 T^{6} - 210 T^{7} + 324 T^{8} - 486 T^{9} + 648 T^{10} - 486 T^{11} + 729 T^{12}$$
$5$ $$1 - 2 T + 11 T^{2} - 26 T^{3} + 88 T^{4} - 150 T^{5} + 509 T^{6} - 750 T^{7} + 2200 T^{8} - 3250 T^{9} + 6875 T^{10} - 6250 T^{11} + 15625 T^{12}$$
$7$ $$( 1 + T )^{6}$$
$11$ $$1 + 2 T + 24 T^{2} + 14 T^{3} + 335 T^{4} - 108 T^{5} + 3520 T^{6} - 1188 T^{7} + 40535 T^{8} + 18634 T^{9} + 351384 T^{10} + 322102 T^{11} + 1771561 T^{12}$$
$13$ 1
$17$ $$1 - 4 T + 63 T^{2} - 172 T^{3} + 1859 T^{4} - 3792 T^{5} + 36442 T^{6} - 64464 T^{7} + 537251 T^{8} - 845036 T^{9} + 5261823 T^{10} - 5679428 T^{11} + 24137569 T^{12}$$
$19$ $$1 - 4 T + 22 T^{2} - 76 T^{3} + 971 T^{4} - 2824 T^{5} + 15500 T^{6} - 53656 T^{7} + 350531 T^{8} - 521284 T^{9} + 2867062 T^{10} - 9904396 T^{11} + 47045881 T^{12}$$
$23$ $$1 + 6 T + 70 T^{2} + 202 T^{3} + 1572 T^{4} + 782 T^{5} + 25728 T^{6} + 17986 T^{7} + 831588 T^{8} + 2457734 T^{9} + 19588870 T^{10} + 38618058 T^{11} + 148035889 T^{12}$$
$29$ $$1 - 10 T + 135 T^{2} - 898 T^{3} + 7715 T^{4} - 39852 T^{5} + 268810 T^{6} - 1155708 T^{7} + 6488315 T^{8} - 21901322 T^{9} + 95482935 T^{10} - 205111490 T^{11} + 594823321 T^{12}$$
$31$ $$1 - 2 T + 144 T^{2} - 358 T^{3} + 9287 T^{4} - 23652 T^{5} + 358576 T^{6} - 733212 T^{7} + 8924807 T^{8} - 10665178 T^{9} + 132987024 T^{10} - 57258302 T^{11} + 887503681 T^{12}$$
$37$ $$1 + 67 T^{2} + 8 T^{3} + 3347 T^{4} + 280 T^{5} + 165506 T^{6} + 10360 T^{7} + 4582043 T^{8} + 405224 T^{9} + 125568787 T^{10} + 2565726409 T^{12}$$
$41$ $$1 + 6 T + 151 T^{2} + 878 T^{3} + 11883 T^{4} + 57724 T^{5} + 604554 T^{6} + 2366684 T^{7} + 19975323 T^{8} + 60512638 T^{9} + 426689911 T^{10} + 695137206 T^{11} + 4750104241 T^{12}$$
$43$ $$1 - 26 T + 472 T^{2} - 5942 T^{3} + 61823 T^{4} - 518084 T^{5} + 3733280 T^{6} - 22277612 T^{7} + 114310727 T^{8} - 472430594 T^{9} + 1613674072 T^{10} - 3822219518 T^{11} + 6321363049 T^{12}$$
$47$ $$1 - 8 T + 178 T^{2} - 792 T^{3} + 11679 T^{4} - 26000 T^{5} + 524700 T^{6} - 1222000 T^{7} + 25798911 T^{8} - 82227816 T^{9} + 868583218 T^{10} - 1834760056 T^{11} + 10779215329 T^{12}$$
$53$ $$1 - 18 T + 267 T^{2} - 3114 T^{3} + 31467 T^{4} - 273420 T^{5} + 2178178 T^{6} - 14491260 T^{7} + 88390803 T^{8} - 463602978 T^{9} + 2106758427 T^{10} - 7527518874 T^{11} + 22164361129 T^{12}$$
$59$ $$1 + 2 T + 272 T^{2} + 506 T^{3} + 32980 T^{4} + 54774 T^{5} + 2408732 T^{6} + 3231666 T^{7} + 114803380 T^{8} + 103921774 T^{9} + 3295922192 T^{10} + 1429848598 T^{11} + 42180533641 T^{12}$$
$61$ $$1 - 28 T + 435 T^{2} - 3956 T^{3} + 18488 T^{4} + 18948 T^{5} - 811259 T^{6} + 1155828 T^{7} + 68793848 T^{8} - 897936836 T^{9} + 6022940835 T^{10} - 23648696428 T^{11} + 51520374361 T^{12}$$
$67$ $$1 + 6 T + 280 T^{2} + 1562 T^{3} + 34127 T^{4} + 179068 T^{5} + 2660672 T^{6} + 11997556 T^{7} + 153196103 T^{8} + 469791806 T^{9} + 5642313880 T^{10} + 8100750642 T^{11} + 90458382169 T^{12}$$
$71$ $$1 + 4 T + 272 T^{2} + 880 T^{3} + 36280 T^{4} + 92988 T^{5} + 3097886 T^{6} + 6602148 T^{7} + 182887480 T^{8} + 314961680 T^{9} + 6911977232 T^{10} + 7216917404 T^{11} + 128100283921 T^{12}$$
$73$ $$1 - 22 T + 375 T^{2} - 3350 T^{3} + 25043 T^{4} - 71940 T^{5} + 489610 T^{6} - 5251620 T^{7} + 133454147 T^{8} - 1303206950 T^{9} + 10649340375 T^{10} - 45607575046 T^{11} + 151334226289 T^{12}$$
$79$ $$1 - 22 T + 536 T^{2} - 7106 T^{3} + 100727 T^{4} - 974028 T^{5} + 10218624 T^{6} - 76948212 T^{7} + 628637207 T^{8} - 3503535134 T^{9} + 20877243416 T^{10} - 67695240778 T^{11} + 243087455521 T^{12}$$
$83$ $$1 - 10 T + 316 T^{2} - 2478 T^{3} + 41691 T^{4} - 288388 T^{5} + 3740088 T^{6} - 23936204 T^{7} + 287209299 T^{8} - 1416888186 T^{9} + 14996829436 T^{10} - 39390406430 T^{11} + 326940373369 T^{12}$$
$89$ $$1 - 4 T + 334 T^{2} - 1140 T^{3} + 57807 T^{4} - 169576 T^{5} + 6306852 T^{6} - 15092264 T^{7} + 457889247 T^{8} - 803664660 T^{9} + 20955908494 T^{10} - 22336237796 T^{11} + 496981290961 T^{12}$$
$97$ $$1 - 12 T + 358 T^{2} - 3260 T^{3} + 59423 T^{4} - 424120 T^{5} + 6536564 T^{6} - 41139640 T^{7} + 559111007 T^{8} - 2975313980 T^{9} + 31693482598 T^{10} - 103048083084 T^{11} + 832972004929 T^{12}$$