# Properties

 Label 38.4.a.b Level $38$ Weight $4$ Character orbit 38.a Self dual yes Analytic conductor $2.242$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$38 = 2 \cdot 19$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 38.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$2.24207258022$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{177})$$ Defining polynomial: $$x^{2} - x - 44$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{177})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -2 q^{2} + ( 1 - \beta ) q^{3} + 4 q^{4} + ( 4 + 2 \beta ) q^{5} + ( -2 + 2 \beta ) q^{6} + ( 29 - \beta ) q^{7} -8 q^{8} + ( 18 - \beta ) q^{9} +O(q^{10})$$ $$q -2 q^{2} + ( 1 - \beta ) q^{3} + 4 q^{4} + ( 4 + 2 \beta ) q^{5} + ( -2 + 2 \beta ) q^{6} + ( 29 - \beta ) q^{7} -8 q^{8} + ( 18 - \beta ) q^{9} + ( -8 - 4 \beta ) q^{10} + ( 6 - 2 \beta ) q^{11} + ( 4 - 4 \beta ) q^{12} + ( 3 + 7 \beta ) q^{13} + ( -58 + 2 \beta ) q^{14} + ( -84 - 4 \beta ) q^{15} + 16 q^{16} + ( -33 + 15 \beta ) q^{17} + ( -36 + 2 \beta ) q^{18} -19 q^{19} + ( 16 + 8 \beta ) q^{20} + ( 73 - 29 \beta ) q^{21} + ( -12 + 4 \beta ) q^{22} + ( -71 - 13 \beta ) q^{23} + ( -8 + 8 \beta ) q^{24} + ( 67 + 20 \beta ) q^{25} + ( -6 - 14 \beta ) q^{26} + ( 35 + 9 \beta ) q^{27} + ( 116 - 4 \beta ) q^{28} + ( -25 - 29 \beta ) q^{29} + ( 168 + 8 \beta ) q^{30} + ( -16 + 16 \beta ) q^{31} -32 q^{32} + ( 94 - 6 \beta ) q^{33} + ( 66 - 30 \beta ) q^{34} + ( 28 + 52 \beta ) q^{35} + ( 72 - 4 \beta ) q^{36} + ( 182 + 16 \beta ) q^{37} + 38 q^{38} + ( -305 - 3 \beta ) q^{39} + ( -32 - 16 \beta ) q^{40} + ( -392 - 6 \beta ) q^{41} + ( -146 + 58 \beta ) q^{42} + ( 172 - 48 \beta ) q^{43} + ( 24 - 8 \beta ) q^{44} + ( -16 + 30 \beta ) q^{45} + ( 142 + 26 \beta ) q^{46} + ( -80 - 40 \beta ) q^{47} + ( 16 - 16 \beta ) q^{48} + ( 542 - 57 \beta ) q^{49} + ( -134 - 40 \beta ) q^{50} + ( -693 + 33 \beta ) q^{51} + ( 12 + 28 \beta ) q^{52} + ( 203 - 9 \beta ) q^{53} + ( -70 - 18 \beta ) q^{54} -152 q^{55} + ( -232 + 8 \beta ) q^{56} + ( -19 + 19 \beta ) q^{57} + ( 50 + 58 \beta ) q^{58} + ( 65 + 71 \beta ) q^{59} + ( -336 - 16 \beta ) q^{60} + ( -318 - 44 \beta ) q^{61} + ( 32 - 32 \beta ) q^{62} + ( 566 - 46 \beta ) q^{63} + 64 q^{64} + ( 628 + 48 \beta ) q^{65} + ( -188 + 12 \beta ) q^{66} + ( -491 + 43 \beta ) q^{67} + ( -132 + 60 \beta ) q^{68} + ( 501 + 71 \beta ) q^{69} + ( -56 - 104 \beta ) q^{70} + ( 214 - 22 \beta ) q^{71} + ( -144 + 8 \beta ) q^{72} + ( 61 + \beta ) q^{73} + ( -364 - 32 \beta ) q^{74} + ( -813 - 67 \beta ) q^{75} -76 q^{76} + ( 262 - 62 \beta ) q^{77} + ( 610 + 6 \beta ) q^{78} + ( 82 - 58 \beta ) q^{79} + ( 64 + 32 \beta ) q^{80} + ( -847 - 8 \beta ) q^{81} + ( 784 + 12 \beta ) q^{82} + ( 1110 + 6 \beta ) q^{83} + ( 292 - 116 \beta ) q^{84} + ( 1188 + 24 \beta ) q^{85} + ( -344 + 96 \beta ) q^{86} + ( 1251 + 25 \beta ) q^{87} + ( -48 + 16 \beta ) q^{88} + ( -440 + 10 \beta ) q^{89} + ( 32 - 60 \beta ) q^{90} + ( -221 + 193 \beta ) q^{91} + ( -284 - 52 \beta ) q^{92} + ( -720 + 16 \beta ) q^{93} + ( 160 + 80 \beta ) q^{94} + ( -76 - 38 \beta ) q^{95} + ( -32 + 32 \beta ) q^{96} + ( -970 + 76 \beta ) q^{97} + ( -1084 + 114 \beta ) q^{98} + ( 196 - 40 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 4q^{2} + q^{3} + 8q^{4} + 10q^{5} - 2q^{6} + 57q^{7} - 16q^{8} + 35q^{9} + O(q^{10})$$ $$2q - 4q^{2} + q^{3} + 8q^{4} + 10q^{5} - 2q^{6} + 57q^{7} - 16q^{8} + 35q^{9} - 20q^{10} + 10q^{11} + 4q^{12} + 13q^{13} - 114q^{14} - 172q^{15} + 32q^{16} - 51q^{17} - 70q^{18} - 38q^{19} + 40q^{20} + 117q^{21} - 20q^{22} - 155q^{23} - 8q^{24} + 154q^{25} - 26q^{26} + 79q^{27} + 228q^{28} - 79q^{29} + 344q^{30} - 16q^{31} - 64q^{32} + 182q^{33} + 102q^{34} + 108q^{35} + 140q^{36} + 380q^{37} + 76q^{38} - 613q^{39} - 80q^{40} - 790q^{41} - 234q^{42} + 296q^{43} + 40q^{44} - 2q^{45} + 310q^{46} - 200q^{47} + 16q^{48} + 1027q^{49} - 308q^{50} - 1353q^{51} + 52q^{52} + 397q^{53} - 158q^{54} - 304q^{55} - 456q^{56} - 19q^{57} + 158q^{58} + 201q^{59} - 688q^{60} - 680q^{61} + 32q^{62} + 1086q^{63} + 128q^{64} + 1304q^{65} - 364q^{66} - 939q^{67} - 204q^{68} + 1073q^{69} - 216q^{70} + 406q^{71} - 280q^{72} + 123q^{73} - 760q^{74} - 1693q^{75} - 152q^{76} + 462q^{77} + 1226q^{78} + 106q^{79} + 160q^{80} - 1702q^{81} + 1580q^{82} + 2226q^{83} + 468q^{84} + 2400q^{85} - 592q^{86} + 2527q^{87} - 80q^{88} - 870q^{89} + 4q^{90} - 249q^{91} - 620q^{92} - 1424q^{93} + 400q^{94} - 190q^{95} - 32q^{96} - 1864q^{97} - 2054q^{98} + 352q^{99} + O(q^{100})$$

## 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
 7.15207 −6.15207
−2.00000 −6.15207 4.00000 18.3041 12.3041 21.8479 −8.00000 10.8479 −36.6083
1.2 −2.00000 7.15207 4.00000 −8.30413 −14.3041 35.1521 −8.00000 24.1521 16.6083
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$19$$ $$1$$

## 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 38.4.a.b 2
3.b odd 2 1 342.4.a.k 2
4.b odd 2 1 304.4.a.d 2
5.b even 2 1 950.4.a.h 2
5.c odd 4 2 950.4.b.g 4
7.b odd 2 1 1862.4.a.b 2
8.b even 2 1 1216.4.a.j 2
8.d odd 2 1 1216.4.a.l 2
19.b odd 2 1 722.4.a.i 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.4.a.b 2 1.a even 1 1 trivial
304.4.a.d 2 4.b odd 2 1
342.4.a.k 2 3.b odd 2 1
722.4.a.i 2 19.b odd 2 1
950.4.a.h 2 5.b even 2 1
950.4.b.g 4 5.c odd 4 2
1216.4.a.j 2 8.b even 2 1
1216.4.a.l 2 8.d odd 2 1
1862.4.a.b 2 7.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - T_{3} - 44$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(38))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + 2 T )^{2}$$
$3$ $$1 - T + 10 T^{2} - 27 T^{3} + 729 T^{4}$$
$5$ $$1 - 10 T + 98 T^{2} - 1250 T^{3} + 15625 T^{4}$$
$7$ $$1 - 57 T + 1454 T^{2} - 19551 T^{3} + 117649 T^{4}$$
$11$ $$1 - 10 T + 2510 T^{2} - 13310 T^{3} + 1771561 T^{4}$$
$13$ $$1 - 13 T + 2268 T^{2} - 28561 T^{3} + 4826809 T^{4}$$
$17$ $$1 + 51 T + 520 T^{2} + 250563 T^{3} + 24137569 T^{4}$$
$19$ $$( 1 + 19 T )^{2}$$
$23$ $$1 + 155 T + 22862 T^{2} + 1885885 T^{3} + 148035889 T^{4}$$
$29$ $$1 + 79 T + 13124 T^{2} + 1926731 T^{3} + 594823321 T^{4}$$
$31$ $$1 + 16 T + 48318 T^{2} + 476656 T^{3} + 887503681 T^{4}$$
$37$ $$1 - 380 T + 126078 T^{2} - 19248140 T^{3} + 2565726409 T^{4}$$
$41$ $$1 + 790 T + 292274 T^{2} + 54447590 T^{3} + 4750104241 T^{4}$$
$43$ $$1 - 296 T + 78966 T^{2} - 23534072 T^{3} + 6321363049 T^{4}$$
$47$ $$1 + 200 T + 146846 T^{2} + 20764600 T^{3} + 10779215329 T^{4}$$
$53$ $$1 - 397 T + 333572 T^{2} - 59104169 T^{3} + 22164361129 T^{4}$$
$59$ $$1 - 201 T + 197794 T^{2} - 41281179 T^{3} + 42180533641 T^{4}$$
$61$ $$1 + 680 T + 483894 T^{2} + 154347080 T^{3} + 51520374361 T^{4}$$
$67$ $$1 + 939 T + 740138 T^{2} + 282416457 T^{3} + 90458382169 T^{4}$$
$71$ $$1 - 406 T + 735614 T^{2} - 145311866 T^{3} + 128100283921 T^{4}$$
$73$ $$1 - 123 T + 781772 T^{2} - 47849091 T^{3} + 151334226289 T^{4}$$
$79$ $$1 - 106 T + 840030 T^{2} - 52262134 T^{3} + 243087455521 T^{4}$$
$83$ $$1 - 2226 T + 2380750 T^{2} - 1272797862 T^{3} + 326940373369 T^{4}$$
$89$ $$1 + 870 T + 1594738 T^{2} + 613323030 T^{3} + 496981290961 T^{4}$$
$97$ $$1 + 1864 T + 2438382 T^{2} + 1701222472 T^{3} + 832972004929 T^{4}$$