# Properties

 Label 2366.2.d.p Level 2366 Weight 2 Character orbit 2366.d Analytic conductor 18.893 Analytic rank 0 Dimension 6 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2366 = 2 \cdot 7 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2366.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$18.8926051182$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.153664.1 Defining polynomial: $$x^{6} + 5 x^{4} + 6 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

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

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + 2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} + 3 \nu$$ $$\beta_{4}$$ $$=$$ $$\nu^{4} + 3 \nu^{2} + 1$$ $$\beta_{5}$$ $$=$$ $$\nu^{5} + 4 \nu^{3} + 3 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - 2$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} - 3 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{4} - 3 \beta_{2} + 5$$ $$\nu^{5}$$ $$=$$ $$\beta_{5} - 4 \beta_{3} + 9 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2366\mathbb{Z}\right)^\times$$.

 $$n$$ $$339$$ $$2199$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
337.1
 1.80194i 0.445042i − 1.24698i − 1.80194i − 0.445042i 1.24698i
1.00000i −0.801938 −1.00000 3.04892i 0.801938i 1.00000i 1.00000i −2.35690 3.04892
337.2 1.00000i 0.554958 −1.00000 1.35690i 0.554958i 1.00000i 1.00000i −2.69202 −1.35690
337.3 1.00000i 2.24698 −1.00000 1.69202i 2.24698i 1.00000i 1.00000i 2.04892 −1.69202
337.4 1.00000i −0.801938 −1.00000 3.04892i 0.801938i 1.00000i 1.00000i −2.35690 3.04892
337.5 1.00000i 0.554958 −1.00000 1.35690i 0.554958i 1.00000i 1.00000i −2.69202 −1.35690
337.6 1.00000i 2.24698 −1.00000 1.69202i 2.24698i 1.00000i 1.00000i 2.04892 −1.69202
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 337.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2366.2.d.p 6
13.b even 2 1 inner 2366.2.d.p 6
13.d odd 4 1 2366.2.a.y 3
13.d odd 4 1 2366.2.a.bd yes 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2366.2.a.y 3 13.d odd 4 1
2366.2.a.bd yes 3 13.d odd 4 1
2366.2.d.p 6 1.a even 1 1 trivial
2366.2.d.p 6 13.b even 2 1 inner

## Hecke kernels

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

 $$T_{3}^{3} - 2 T_{3}^{2} - T_{3} + 1$$ $$T_{5}^{6} + 14 T_{5}^{4} + 49 T_{5}^{2} + 49$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{3}$$
$3$ $$( 1 - 2 T + 8 T^{2} - 11 T^{3} + 24 T^{4} - 18 T^{5} + 27 T^{6} )^{2}$$
$5$ $$1 - 16 T^{2} + 144 T^{4} - 841 T^{6} + 3600 T^{8} - 10000 T^{10} + 15625 T^{12}$$
$7$ $$( 1 + T^{2} )^{3}$$
$11$ $$1 - 61 T^{2} + 1601 T^{4} - 23121 T^{6} + 193721 T^{8} - 893101 T^{10} + 1771561 T^{12}$$
$13$ 1
$17$ $$( 1 + 2 T + 36 T^{2} + 81 T^{3} + 612 T^{4} + 578 T^{5} + 4913 T^{6} )^{2}$$
$19$ $$1 - 25 T^{2} + 785 T^{4} - 18609 T^{6} + 283385 T^{8} - 3258025 T^{10} + 47045881 T^{12}$$
$23$ $$( 1 + T + 53 T^{2} + 59 T^{3} + 1219 T^{4} + 529 T^{5} + 12167 T^{6} )^{2}$$
$29$ $$( 1 - 5 T + 65 T^{2} - 193 T^{3} + 1885 T^{4} - 4205 T^{5} + 24389 T^{6} )^{2}$$
$31$ $$1 - 92 T^{2} + 3952 T^{4} - 126101 T^{6} + 3797872 T^{8} - 84963932 T^{10} + 887503681 T^{12}$$
$37$ $$1 + 44 T^{2} + 3952 T^{4} + 111455 T^{6} + 5410288 T^{8} + 82463084 T^{10} + 2565726409 T^{12}$$
$41$ $$1 - 129 T^{2} + 6677 T^{4} - 251657 T^{6} + 11224037 T^{8} - 364523169 T^{10} + 4750104241 T^{12}$$
$43$ $$( 1 - T + 85 T^{2} - 169 T^{3} + 3655 T^{4} - 1849 T^{5} + 79507 T^{6} )^{2}$$
$47$ $$1 - 37 T^{2} + 5597 T^{4} - 164529 T^{6} + 12363773 T^{8} - 180548197 T^{10} + 10779215329 T^{12}$$
$53$ $$( 1 + 12 T + 186 T^{2} + 1259 T^{3} + 9858 T^{4} + 33708 T^{5} + 148877 T^{6} )^{2}$$
$59$ $$1 - 225 T^{2} + 24609 T^{4} - 1732041 T^{6} + 85663929 T^{8} - 2726406225 T^{10} + 42180533641 T^{12}$$
$61$ $$( 1 + 3 T + 95 T^{2} + 479 T^{3} + 5795 T^{4} + 11163 T^{5} + 226981 T^{6} )^{2}$$
$67$ $$1 - 277 T^{2} + 37585 T^{4} - 3135385 T^{6} + 168719065 T^{8} - 5581860517 T^{10} + 90458382169 T^{12}$$
$71$ $$1 - 328 T^{2} + 50184 T^{4} - 4526773 T^{6} + 252977544 T^{8} - 8335031368 T^{10} + 128100283921 T^{12}$$
$73$ $$1 + 143 T^{2} + 14261 T^{4} + 1180647 T^{6} + 75996869 T^{8} + 4060948463 T^{10} + 151334226289 T^{12}$$
$79$ $$( 1 + 2 T + 12 T^{2} - 931 T^{3} + 948 T^{4} + 12482 T^{5} + 493039 T^{6} )^{2}$$
$83$ $$1 - 457 T^{2} + 90197 T^{4} - 9818049 T^{6} + 621367133 T^{8} - 21688452697 T^{10} + 326940373369 T^{12}$$
$89$ $$1 - 156 T^{2} + 956 T^{4} + 981847 T^{6} + 7572476 T^{8} - 9787789596 T^{10} + 496981290961 T^{12}$$
$97$ $$1 - 493 T^{2} + 108317 T^{4} - 13552161 T^{6} + 1019154653 T^{8} - 43644935533 T^{10} + 832972004929 T^{12}$$