# Properties

 Label 38.4.c.a Level 38 Weight 4 Character orbit 38.c Analytic conductor 2.242 Analytic rank 0 Dimension 2 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$38 = 2 \cdot 19$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 38.c (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.24207258022$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -2 + 2 \zeta_{6} ) q^{2} + ( -5 + 5 \zeta_{6} ) q^{3} -4 \zeta_{6} q^{4} + ( -3 + 3 \zeta_{6} ) q^{5} -10 \zeta_{6} q^{6} -32 q^{7} + 8 q^{8} + 2 \zeta_{6} q^{9} +O(q^{10})$$ $$q + ( -2 + 2 \zeta_{6} ) q^{2} + ( -5 + 5 \zeta_{6} ) q^{3} -4 \zeta_{6} q^{4} + ( -3 + 3 \zeta_{6} ) q^{5} -10 \zeta_{6} q^{6} -32 q^{7} + 8 q^{8} + 2 \zeta_{6} q^{9} -6 \zeta_{6} q^{10} + 4 q^{11} + 20 q^{12} + 69 \zeta_{6} q^{13} + ( 64 - 64 \zeta_{6} ) q^{14} -15 \zeta_{6} q^{15} + ( -16 + 16 \zeta_{6} ) q^{16} + ( -19 + 19 \zeta_{6} ) q^{17} -4 q^{18} + ( 57 + 38 \zeta_{6} ) q^{19} + 12 q^{20} + ( 160 - 160 \zeta_{6} ) q^{21} + ( -8 + 8 \zeta_{6} ) q^{22} -67 \zeta_{6} q^{23} + ( -40 + 40 \zeta_{6} ) q^{24} + 116 \zeta_{6} q^{25} -138 q^{26} -145 q^{27} + 128 \zeta_{6} q^{28} -51 \zeta_{6} q^{29} + 30 q^{30} -132 q^{31} -32 \zeta_{6} q^{32} + ( -20 + 20 \zeta_{6} ) q^{33} -38 \zeta_{6} q^{34} + ( 96 - 96 \zeta_{6} ) q^{35} + ( 8 - 8 \zeta_{6} ) q^{36} -14 q^{37} + ( -190 + 114 \zeta_{6} ) q^{38} -345 q^{39} + ( -24 + 24 \zeta_{6} ) q^{40} + ( 413 - 413 \zeta_{6} ) q^{41} + 320 \zeta_{6} q^{42} + ( -129 + 129 \zeta_{6} ) q^{43} -16 \zeta_{6} q^{44} -6 q^{45} + 134 q^{46} + 617 \zeta_{6} q^{47} -80 \zeta_{6} q^{48} + 681 q^{49} -232 q^{50} -95 \zeta_{6} q^{51} + ( 276 - 276 \zeta_{6} ) q^{52} -383 \zeta_{6} q^{53} + ( 290 - 290 \zeta_{6} ) q^{54} + ( -12 + 12 \zeta_{6} ) q^{55} -256 q^{56} + ( -475 + 285 \zeta_{6} ) q^{57} + 102 q^{58} + ( 599 - 599 \zeta_{6} ) q^{59} + ( -60 + 60 \zeta_{6} ) q^{60} + 217 \zeta_{6} q^{61} + ( 264 - 264 \zeta_{6} ) q^{62} -64 \zeta_{6} q^{63} + 64 q^{64} -207 q^{65} -40 \zeta_{6} q^{66} + 225 \zeta_{6} q^{67} + 76 q^{68} + 335 q^{69} + 192 \zeta_{6} q^{70} + ( -701 + 701 \zeta_{6} ) q^{71} + 16 \zeta_{6} q^{72} + ( -1015 + 1015 \zeta_{6} ) q^{73} + ( 28 - 28 \zeta_{6} ) q^{74} -580 q^{75} + ( 152 - 380 \zeta_{6} ) q^{76} -128 q^{77} + ( 690 - 690 \zeta_{6} ) q^{78} + ( -349 + 349 \zeta_{6} ) q^{79} -48 \zeta_{6} q^{80} + ( 671 - 671 \zeta_{6} ) q^{81} + 826 \zeta_{6} q^{82} -592 q^{83} -640 q^{84} -57 \zeta_{6} q^{85} -258 \zeta_{6} q^{86} + 255 q^{87} + 32 q^{88} + 1349 \zeta_{6} q^{89} + ( 12 - 12 \zeta_{6} ) q^{90} -2208 \zeta_{6} q^{91} + ( -268 + 268 \zeta_{6} ) q^{92} + ( 660 - 660 \zeta_{6} ) q^{93} -1234 q^{94} + ( -285 + 171 \zeta_{6} ) q^{95} + 160 q^{96} + ( 613 - 613 \zeta_{6} ) q^{97} + ( -1362 + 1362 \zeta_{6} ) q^{98} + 8 \zeta_{6} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} - 5q^{3} - 4q^{4} - 3q^{5} - 10q^{6} - 64q^{7} + 16q^{8} + 2q^{9} + O(q^{10})$$ $$2q - 2q^{2} - 5q^{3} - 4q^{4} - 3q^{5} - 10q^{6} - 64q^{7} + 16q^{8} + 2q^{9} - 6q^{10} + 8q^{11} + 40q^{12} + 69q^{13} + 64q^{14} - 15q^{15} - 16q^{16} - 19q^{17} - 8q^{18} + 152q^{19} + 24q^{20} + 160q^{21} - 8q^{22} - 67q^{23} - 40q^{24} + 116q^{25} - 276q^{26} - 290q^{27} + 128q^{28} - 51q^{29} + 60q^{30} - 264q^{31} - 32q^{32} - 20q^{33} - 38q^{34} + 96q^{35} + 8q^{36} - 28q^{37} - 266q^{38} - 690q^{39} - 24q^{40} + 413q^{41} + 320q^{42} - 129q^{43} - 16q^{44} - 12q^{45} + 268q^{46} + 617q^{47} - 80q^{48} + 1362q^{49} - 464q^{50} - 95q^{51} + 276q^{52} - 383q^{53} + 290q^{54} - 12q^{55} - 512q^{56} - 665q^{57} + 204q^{58} + 599q^{59} - 60q^{60} + 217q^{61} + 264q^{62} - 64q^{63} + 128q^{64} - 414q^{65} - 40q^{66} + 225q^{67} + 152q^{68} + 670q^{69} + 192q^{70} - 701q^{71} + 16q^{72} - 1015q^{73} + 28q^{74} - 1160q^{75} - 76q^{76} - 256q^{77} + 690q^{78} - 349q^{79} - 48q^{80} + 671q^{81} + 826q^{82} - 1184q^{83} - 1280q^{84} - 57q^{85} - 258q^{86} + 510q^{87} + 64q^{88} + 1349q^{89} + 12q^{90} - 2208q^{91} - 268q^{92} + 660q^{93} - 2468q^{94} - 399q^{95} + 320q^{96} + 613q^{97} - 1362q^{98} + 8q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/38\mathbb{Z}\right)^\times$$.

 $$n$$ $$21$$ $$\chi(n)$$ $$-\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1
 0.5 − 0.866025i 0.5 + 0.866025i
−1.00000 1.73205i −2.50000 4.33013i −2.00000 + 3.46410i −1.50000 2.59808i −5.00000 + 8.66025i −32.0000 8.00000 1.00000 1.73205i −3.00000 + 5.19615i
11.1 −1.00000 + 1.73205i −2.50000 + 4.33013i −2.00000 3.46410i −1.50000 + 2.59808i −5.00000 8.66025i −32.0000 8.00000 1.00000 + 1.73205i −3.00000 5.19615i
 $$n$$: 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
19.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 38.4.c.a 2
3.b odd 2 1 342.4.g.d 2
4.b odd 2 1 304.4.i.b 2
19.c even 3 1 inner 38.4.c.a 2
19.c even 3 1 722.4.a.e 1
19.d odd 6 1 722.4.a.a 1
57.h odd 6 1 342.4.g.d 2
76.g odd 6 1 304.4.i.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.4.c.a 2 1.a even 1 1 trivial
38.4.c.a 2 19.c even 3 1 inner
304.4.i.b 2 4.b odd 2 1
304.4.i.b 2 76.g odd 6 1
342.4.g.d 2 3.b odd 2 1
342.4.g.d 2 57.h odd 6 1
722.4.a.a 1 19.d odd 6 1
722.4.a.e 1 19.c even 3 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 5 T_{3} + 25$$ acting on $$S_{4}^{\mathrm{new}}(38, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 2 T + 4 T^{2}$$
$3$ $$1 + 5 T - 2 T^{2} + 135 T^{3} + 729 T^{4}$$
$5$ $$1 + 3 T - 116 T^{2} + 375 T^{3} + 15625 T^{4}$$
$7$ $$( 1 + 32 T + 343 T^{2} )^{2}$$
$11$ $$( 1 - 4 T + 1331 T^{2} )^{2}$$
$13$ $$1 - 69 T + 2564 T^{2} - 151593 T^{3} + 4826809 T^{4}$$
$17$ $$1 + 19 T - 4552 T^{2} + 93347 T^{3} + 24137569 T^{4}$$
$19$ $$1 - 152 T + 6859 T^{2}$$
$23$ $$1 + 67 T - 7678 T^{2} + 815189 T^{3} + 148035889 T^{4}$$
$29$ $$1 + 51 T - 21788 T^{2} + 1243839 T^{3} + 594823321 T^{4}$$
$31$ $$( 1 + 132 T + 29791 T^{2} )^{2}$$
$37$ $$( 1 + 14 T + 50653 T^{2} )^{2}$$
$41$ $$1 - 413 T + 101648 T^{2} - 28464373 T^{3} + 4750104241 T^{4}$$
$43$ $$1 + 129 T - 62866 T^{2} + 10256403 T^{3} + 6321363049 T^{4}$$
$47$ $$1 - 617 T + 276866 T^{2} - 64058791 T^{3} + 10779215329 T^{4}$$
$53$ $$1 + 383 T - 2188 T^{2} + 57019891 T^{3} + 22164361129 T^{4}$$
$59$ $$1 - 599 T + 153422 T^{2} - 123022021 T^{3} + 42180533641 T^{4}$$
$61$ $$1 - 217 T - 179892 T^{2} - 49254877 T^{3} + 51520374361 T^{4}$$
$67$ $$1 - 225 T - 250138 T^{2} - 67671675 T^{3} + 90458382169 T^{4}$$
$71$ $$1 + 701 T + 133490 T^{2} + 250895611 T^{3} + 128100283921 T^{4}$$
$73$ $$1 + 1015 T + 641208 T^{2} + 394852255 T^{3} + 151334226289 T^{4}$$
$79$ $$1 + 349 T - 371238 T^{2} + 172070611 T^{3} + 243087455521 T^{4}$$
$83$ $$( 1 + 592 T + 571787 T^{2} )^{2}$$
$89$ $$1 - 1349 T + 1114832 T^{2} - 951003181 T^{3} + 496981290961 T^{4}$$
$97$ $$1 - 613 T - 536904 T^{2} - 559468549 T^{3} + 832972004929 T^{4}$$