# Properties

 Label 2366.2.d.f Level 2366 Weight 2 Character orbit 2366.d Analytic conductor 18.893 Analytic rank 0 Dimension 2 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: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 182) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 − 1.00000i 1.00000i
1.00000i 1.00000 −1.00000 3.00000i 1.00000i 1.00000i 1.00000i −2.00000 3.00000
337.2 1.00000i 1.00000 −1.00000 3.00000i 1.00000i 1.00000i 1.00000i −2.00000 3.00000
 $$n$$: e.g. 2-40 or 990-1000 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.f 2
13.b even 2 1 inner 2366.2.d.f 2
13.d odd 4 1 2366.2.a.f 1
13.d odd 4 1 2366.2.a.n 1
13.f odd 12 2 182.2.g.b 2
39.k even 12 2 1638.2.r.c 2
52.l even 12 2 1456.2.s.e 2
91.w even 12 2 1274.2.h.i 2
91.x odd 12 2 1274.2.e.d 2
91.ba even 12 2 1274.2.e.i 2
91.bc even 12 2 1274.2.g.g 2
91.bd odd 12 2 1274.2.h.j 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
182.2.g.b 2 13.f odd 12 2
1274.2.e.d 2 91.x odd 12 2
1274.2.e.i 2 91.ba even 12 2
1274.2.g.g 2 91.bc even 12 2
1274.2.h.i 2 91.w even 12 2
1274.2.h.j 2 91.bd odd 12 2
1456.2.s.e 2 52.l even 12 2
1638.2.r.c 2 39.k even 12 2
2366.2.a.f 1 13.d odd 4 1
2366.2.a.n 1 13.d odd 4 1
2366.2.d.f 2 1.a even 1 1 trivial
2366.2.d.f 2 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 $$S_{2}^{\mathrm{new}}(2366, [\chi])$$:

 $$T_{3} - 1$$ $$T_{5}^{2} + 9$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2}$$
$3$ $$( 1 - T + 3 T^{2} )^{2}$$
$5$ $$1 - T^{2} + 25 T^{4}$$
$7$ $$1 + T^{2}$$
$11$ $$( 1 - 11 T^{2} )^{2}$$
$13$ 1
$17$ $$( 1 + 6 T + 17 T^{2} )^{2}$$
$19$ $$1 - 22 T^{2} + 361 T^{4}$$
$23$ $$( 1 + 3 T + 23 T^{2} )^{2}$$
$29$ $$( 1 - 6 T + 29 T^{2} )^{2}$$
$31$ $$1 + 38 T^{2} + 961 T^{4}$$
$37$ $$1 - 10 T^{2} + 1369 T^{4}$$
$41$ $$( 1 - 41 T^{2} )^{2}$$
$43$ $$( 1 + 8 T + 43 T^{2} )^{2}$$
$47$ $$1 - 58 T^{2} + 2209 T^{4}$$
$53$ $$( 1 - 12 T + 53 T^{2} )^{2}$$
$59$ $$1 - 109 T^{2} + 3481 T^{4}$$
$61$ $$( 1 - 11 T + 61 T^{2} )^{2}$$
$67$ $$1 - 130 T^{2} + 4489 T^{4}$$
$71$ $$1 - 133 T^{2} + 5041 T^{4}$$
$73$ $$1 - 142 T^{2} + 5329 T^{4}$$
$79$ $$( 1 + 4 T + 79 T^{2} )^{2}$$
$83$ $$( 1 - 83 T^{2} )^{2}$$
$89$ $$1 - 142 T^{2} + 7921 T^{4}$$
$97$ $$1 - 190 T^{2} + 9409 T^{4}$$