# Properties

 Label 38.4.c.b 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.

 $$f(q)$$ $$=$$ $$q + ( -2 + 2 \zeta_{6} ) q^{2} + ( 5 - 5 \zeta_{6} ) q^{3} -4 \zeta_{6} q^{4} + ( 12 - 12 \zeta_{6} ) q^{5} + 10 \zeta_{6} q^{6} + 8 q^{7} + 8 q^{8} + 2 \zeta_{6} q^{9} +O(q^{10})$$ $$q + ( -2 + 2 \zeta_{6} ) q^{2} + ( 5 - 5 \zeta_{6} ) q^{3} -4 \zeta_{6} q^{4} + ( 12 - 12 \zeta_{6} ) q^{5} + 10 \zeta_{6} q^{6} + 8 q^{7} + 8 q^{8} + 2 \zeta_{6} q^{9} + 24 \zeta_{6} q^{10} + 9 q^{11} -20 q^{12} -26 \zeta_{6} q^{13} + ( -16 + 16 \zeta_{6} ) q^{14} -60 \zeta_{6} q^{15} + ( -16 + 16 \zeta_{6} ) q^{16} + ( -114 + 114 \zeta_{6} ) q^{17} -4 q^{18} + ( -38 - 57 \zeta_{6} ) q^{19} -48 q^{20} + ( 40 - 40 \zeta_{6} ) q^{21} + ( -18 + 18 \zeta_{6} ) q^{22} + 78 \zeta_{6} q^{23} + ( 40 - 40 \zeta_{6} ) q^{24} -19 \zeta_{6} q^{25} + 52 q^{26} + 145 q^{27} -32 \zeta_{6} q^{28} + 204 \zeta_{6} q^{29} + 120 q^{30} + 98 q^{31} -32 \zeta_{6} q^{32} + ( 45 - 45 \zeta_{6} ) q^{33} -228 \zeta_{6} q^{34} + ( 96 - 96 \zeta_{6} ) q^{35} + ( 8 - 8 \zeta_{6} ) q^{36} -334 q^{37} + ( 190 - 76 \zeta_{6} ) q^{38} -130 q^{39} + ( 96 - 96 \zeta_{6} ) q^{40} + ( -177 + 177 \zeta_{6} ) q^{41} + 80 \zeta_{6} q^{42} + ( 316 - 316 \zeta_{6} ) q^{43} -36 \zeta_{6} q^{44} + 24 q^{45} -156 q^{46} + 492 \zeta_{6} q^{47} + 80 \zeta_{6} q^{48} -279 q^{49} + 38 q^{50} + 570 \zeta_{6} q^{51} + ( -104 + 104 \zeta_{6} ) q^{52} -678 \zeta_{6} q^{53} + ( -290 + 290 \zeta_{6} ) q^{54} + ( 108 - 108 \zeta_{6} ) q^{55} + 64 q^{56} + ( -475 + 190 \zeta_{6} ) q^{57} -408 q^{58} + ( 579 - 579 \zeta_{6} ) q^{59} + ( -240 + 240 \zeta_{6} ) q^{60} + 352 \zeta_{6} q^{61} + ( -196 + 196 \zeta_{6} ) q^{62} + 16 \zeta_{6} q^{63} + 64 q^{64} -312 q^{65} + 90 \zeta_{6} q^{66} -755 \zeta_{6} q^{67} + 456 q^{68} + 390 q^{69} + 192 \zeta_{6} q^{70} + ( -6 + 6 \zeta_{6} ) q^{71} + 16 \zeta_{6} q^{72} + ( 145 - 145 \zeta_{6} ) q^{73} + ( 668 - 668 \zeta_{6} ) q^{74} -95 q^{75} + ( -228 + 380 \zeta_{6} ) q^{76} + 72 q^{77} + ( 260 - 260 \zeta_{6} ) q^{78} + ( 316 - 316 \zeta_{6} ) q^{79} + 192 \zeta_{6} q^{80} + ( 671 - 671 \zeta_{6} ) q^{81} -354 \zeta_{6} q^{82} -567 q^{83} -160 q^{84} + 1368 \zeta_{6} q^{85} + 632 \zeta_{6} q^{86} + 1020 q^{87} + 72 q^{88} + 114 \zeta_{6} q^{89} + ( -48 + 48 \zeta_{6} ) q^{90} -208 \zeta_{6} q^{91} + ( 312 - 312 \zeta_{6} ) q^{92} + ( 490 - 490 \zeta_{6} ) q^{93} -984 q^{94} + ( -1140 + 456 \zeta_{6} ) q^{95} -160 q^{96} + ( 943 - 943 \zeta_{6} ) q^{97} + ( 558 - 558 \zeta_{6} ) q^{98} + 18 \zeta_{6} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} + 5q^{3} - 4q^{4} + 12q^{5} + 10q^{6} + 16q^{7} + 16q^{8} + 2q^{9} + O(q^{10})$$ $$2q - 2q^{2} + 5q^{3} - 4q^{4} + 12q^{5} + 10q^{6} + 16q^{7} + 16q^{8} + 2q^{9} + 24q^{10} + 18q^{11} - 40q^{12} - 26q^{13} - 16q^{14} - 60q^{15} - 16q^{16} - 114q^{17} - 8q^{18} - 133q^{19} - 96q^{20} + 40q^{21} - 18q^{22} + 78q^{23} + 40q^{24} - 19q^{25} + 104q^{26} + 290q^{27} - 32q^{28} + 204q^{29} + 240q^{30} + 196q^{31} - 32q^{32} + 45q^{33} - 228q^{34} + 96q^{35} + 8q^{36} - 668q^{37} + 304q^{38} - 260q^{39} + 96q^{40} - 177q^{41} + 80q^{42} + 316q^{43} - 36q^{44} + 48q^{45} - 312q^{46} + 492q^{47} + 80q^{48} - 558q^{49} + 76q^{50} + 570q^{51} - 104q^{52} - 678q^{53} - 290q^{54} + 108q^{55} + 128q^{56} - 760q^{57} - 816q^{58} + 579q^{59} - 240q^{60} + 352q^{61} - 196q^{62} + 16q^{63} + 128q^{64} - 624q^{65} + 90q^{66} - 755q^{67} + 912q^{68} + 780q^{69} + 192q^{70} - 6q^{71} + 16q^{72} + 145q^{73} + 668q^{74} - 190q^{75} - 76q^{76} + 144q^{77} + 260q^{78} + 316q^{79} + 192q^{80} + 671q^{81} - 354q^{82} - 1134q^{83} - 320q^{84} + 1368q^{85} + 632q^{86} + 2040q^{87} + 144q^{88} + 114q^{89} - 48q^{90} - 208q^{91} + 312q^{92} + 490q^{93} - 1968q^{94} - 1824q^{95} - 320q^{96} + 943q^{97} + 558q^{98} + 18q^{99} + O(q^{100})$$

## 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 6.00000 + 10.3923i 5.00000 8.66025i 8.00000 8.00000 1.00000 1.73205i 12.0000 20.7846i
11.1 −1.00000 + 1.73205i 2.50000 4.33013i −2.00000 3.46410i 6.00000 10.3923i 5.00000 + 8.66025i 8.00000 8.00000 1.00000 + 1.73205i 12.0000 + 20.7846i
 $$n$$: 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.b 2
3.b odd 2 1 342.4.g.c 2
4.b odd 2 1 304.4.i.a 2
19.c even 3 1 inner 38.4.c.b 2
19.c even 3 1 722.4.a.c 1
19.d odd 6 1 722.4.a.b 1
57.h odd 6 1 342.4.g.c 2
76.g odd 6 1 304.4.i.a 2

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

$p$ $F_p(T)$
$2$ $$1 + 2 T + 4 T^{2}$$
$3$ $$1 - 5 T - 2 T^{2} - 135 T^{3} + 729 T^{4}$$
$5$ $$1 - 12 T + 19 T^{2} - 1500 T^{3} + 15625 T^{4}$$
$7$ $$( 1 - 8 T + 343 T^{2} )^{2}$$
$11$ $$( 1 - 9 T + 1331 T^{2} )^{2}$$
$13$ $$( 1 - 65 T + 2197 T^{2} )( 1 + 91 T + 2197 T^{2} )$$
$17$ $$1 + 114 T + 8083 T^{2} + 560082 T^{3} + 24137569 T^{4}$$
$19$ $$1 + 133 T + 6859 T^{2}$$
$23$ $$1 - 78 T - 6083 T^{2} - 949026 T^{3} + 148035889 T^{4}$$
$29$ $$1 - 204 T + 17227 T^{2} - 4975356 T^{3} + 594823321 T^{4}$$
$31$ $$( 1 - 98 T + 29791 T^{2} )^{2}$$
$37$ $$( 1 + 334 T + 50653 T^{2} )^{2}$$
$41$ $$1 + 177 T - 37592 T^{2} + 12199017 T^{3} + 4750104241 T^{4}$$
$43$ $$1 - 316 T + 20349 T^{2} - 25124212 T^{3} + 6321363049 T^{4}$$
$47$ $$1 - 492 T + 138241 T^{2} - 51080916 T^{3} + 10779215329 T^{4}$$
$53$ $$1 + 678 T + 310807 T^{2} + 100938606 T^{3} + 22164361129 T^{4}$$
$59$ $$1 - 579 T + 129862 T^{2} - 118914441 T^{3} + 42180533641 T^{4}$$
$61$ $$1 - 352 T - 103077 T^{2} - 79897312 T^{3} + 51520374361 T^{4}$$
$67$ $$1 + 755 T + 269262 T^{2} + 227076065 T^{3} + 90458382169 T^{4}$$
$71$ $$1 + 6 T - 357875 T^{2} + 2147466 T^{3} + 128100283921 T^{4}$$
$73$ $$1 - 145 T - 367992 T^{2} - 56407465 T^{3} + 151334226289 T^{4}$$
$79$ $$( 1 - 1343 T + 493039 T^{2} )( 1 + 1027 T + 493039 T^{2} )$$
$83$ $$( 1 + 567 T + 571787 T^{2} )^{2}$$
$89$ $$1 - 114 T - 691973 T^{2} - 80366466 T^{3} + 496981290961 T^{4}$$
$97$ $$1 - 943 T - 23424 T^{2} - 860650639 T^{3} + 832972004929 T^{4}$$