# Properties

 Label 38.4.a.c 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: $$\Q(\sqrt{73})$$ Defining polynomial: $$x^{2} - x - 18$$ 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 $$\beta = \frac{1}{2}(1 + \sqrt{73})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 q^{2} + ( 5 - \beta ) q^{3} + 4 q^{4} + ( -6 + 3 \beta ) q^{5} + ( 10 - 2 \beta ) q^{6} + ( -11 + 4 \beta ) q^{7} + 8 q^{8} + ( 16 - 9 \beta ) q^{9} +O(q^{10})$$ $$q + 2 q^{2} + ( 5 - \beta ) q^{3} + 4 q^{4} + ( -6 + 3 \beta ) q^{5} + ( 10 - 2 \beta ) q^{6} + ( -11 + 4 \beta ) q^{7} + 8 q^{8} + ( 16 - 9 \beta ) q^{9} + ( -12 + 6 \beta ) q^{10} + ( -8 - \beta ) q^{11} + ( 20 - 4 \beta ) q^{12} + ( 15 - 13 \beta ) q^{13} + ( -22 + 8 \beta ) q^{14} + ( -84 + 18 \beta ) q^{15} + 16 q^{16} + ( -41 + 2 \beta ) q^{17} + ( 32 - 18 \beta ) q^{18} + 19 q^{19} + ( -24 + 12 \beta ) q^{20} + ( -127 + 27 \beta ) q^{21} + ( -16 - 2 \beta ) q^{22} + ( 43 - 13 \beta ) q^{23} + ( 40 - 8 \beta ) q^{24} + ( 73 - 27 \beta ) q^{25} + ( 30 - 26 \beta ) q^{26} + ( 107 - 25 \beta ) q^{27} + ( -44 + 16 \beta ) q^{28} + ( -9 + 21 \beta ) q^{29} + ( -168 + 36 \beta ) q^{30} + ( 84 + 44 \beta ) q^{31} + 32 q^{32} + ( -22 + 4 \beta ) q^{33} + ( -82 + 4 \beta ) q^{34} + ( 282 - 45 \beta ) q^{35} + ( 64 - 36 \beta ) q^{36} + ( 82 + 28 \beta ) q^{37} + 38 q^{38} + ( 309 - 67 \beta ) q^{39} + ( -48 + 24 \beta ) q^{40} + ( -20 - 10 \beta ) q^{41} + ( -254 + 54 \beta ) q^{42} + ( 342 - 7 \beta ) q^{43} + ( -32 - 4 \beta ) q^{44} + ( -582 + 75 \beta ) q^{45} + ( 86 - 26 \beta ) q^{46} + ( -230 + 71 \beta ) q^{47} + ( 80 - 16 \beta ) q^{48} + ( 66 - 72 \beta ) q^{49} + ( 146 - 54 \beta ) q^{50} + ( -241 + 49 \beta ) q^{51} + ( 60 - 52 \beta ) q^{52} + ( -601 - 17 \beta ) q^{53} + ( 214 - 50 \beta ) q^{54} + ( -6 - 21 \beta ) q^{55} + ( -88 + 32 \beta ) q^{56} + ( 95 - 19 \beta ) q^{57} + ( -18 + 42 \beta ) q^{58} + ( -131 - 25 \beta ) q^{59} + ( -336 + 72 \beta ) q^{60} + ( 212 - 111 \beta ) q^{61} + ( 168 + 88 \beta ) q^{62} + ( -824 + 127 \beta ) q^{63} + 64 q^{64} + ( -792 + 84 \beta ) q^{65} + ( -44 + 8 \beta ) q^{66} + ( 573 + 77 \beta ) q^{67} + ( -164 + 8 \beta ) q^{68} + ( 449 - 95 \beta ) q^{69} + ( 564 - 90 \beta ) q^{70} + ( 158 - 116 \beta ) q^{71} + ( 128 - 72 \beta ) q^{72} + ( 97 + 184 \beta ) q^{73} + ( 164 + 56 \beta ) q^{74} + ( 851 - 181 \beta ) q^{75} + 76 q^{76} + ( 16 - 25 \beta ) q^{77} + ( 618 - 134 \beta ) q^{78} + ( 646 + 58 \beta ) q^{79} + ( -96 + 48 \beta ) q^{80} + ( 553 + 36 \beta ) q^{81} + ( -40 - 20 \beta ) q^{82} + ( -238 - 194 \beta ) q^{83} + ( -508 + 108 \beta ) q^{84} + ( 354 - 129 \beta ) q^{85} + ( 684 - 14 \beta ) q^{86} + ( -423 + 93 \beta ) q^{87} + ( -64 - 8 \beta ) q^{88} + ( -212 + 188 \beta ) q^{89} + ( -1164 + 150 \beta ) q^{90} + ( -1101 + 151 \beta ) q^{91} + ( 172 - 52 \beta ) q^{92} + ( -372 + 92 \beta ) q^{93} + ( -460 + 142 \beta ) q^{94} + ( -114 + 57 \beta ) q^{95} + ( 160 - 32 \beta ) q^{96} + ( 698 - 102 \beta ) q^{97} + ( 132 - 144 \beta ) q^{98} + ( 34 + 65 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 4q^{2} + 9q^{3} + 8q^{4} - 9q^{5} + 18q^{6} - 18q^{7} + 16q^{8} + 23q^{9} + O(q^{10})$$ $$2q + 4q^{2} + 9q^{3} + 8q^{4} - 9q^{5} + 18q^{6} - 18q^{7} + 16q^{8} + 23q^{9} - 18q^{10} - 17q^{11} + 36q^{12} + 17q^{13} - 36q^{14} - 150q^{15} + 32q^{16} - 80q^{17} + 46q^{18} + 38q^{19} - 36q^{20} - 227q^{21} - 34q^{22} + 73q^{23} + 72q^{24} + 119q^{25} + 34q^{26} + 189q^{27} - 72q^{28} + 3q^{29} - 300q^{30} + 212q^{31} + 64q^{32} - 40q^{33} - 160q^{34} + 519q^{35} + 92q^{36} + 192q^{37} + 76q^{38} + 551q^{39} - 72q^{40} - 50q^{41} - 454q^{42} + 677q^{43} - 68q^{44} - 1089q^{45} + 146q^{46} - 389q^{47} + 144q^{48} + 60q^{49} + 238q^{50} - 433q^{51} + 68q^{52} - 1219q^{53} + 378q^{54} - 33q^{55} - 144q^{56} + 171q^{57} + 6q^{58} - 287q^{59} - 600q^{60} + 313q^{61} + 424q^{62} - 1521q^{63} + 128q^{64} - 1500q^{65} - 80q^{66} + 1223q^{67} - 320q^{68} + 803q^{69} + 1038q^{70} + 200q^{71} + 184q^{72} + 378q^{73} + 384q^{74} + 1521q^{75} + 152q^{76} + 7q^{77} + 1102q^{78} + 1350q^{79} - 144q^{80} + 1142q^{81} - 100q^{82} - 670q^{83} - 908q^{84} + 579q^{85} + 1354q^{86} - 753q^{87} - 136q^{88} - 236q^{89} - 2178q^{90} - 2051q^{91} + 292q^{92} - 652q^{93} - 778q^{94} - 171q^{95} + 288q^{96} + 1294q^{97} + 120q^{98} + 133q^{99} + O(q^{100})$$

## 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
 4.77200 −3.77200
2.00000 0.227998 4.00000 8.31601 0.455996 8.08801 8.00000 −26.9480 16.6320
1.2 2.00000 8.77200 4.00000 −17.3160 17.5440 −26.0880 8.00000 49.9480 −34.6320
 $$n$$: 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.c 2
3.b odd 2 1 342.4.a.h 2
4.b odd 2 1 304.4.a.c 2
5.b even 2 1 950.4.a.e 2
5.c odd 4 2 950.4.b.i 4
7.b odd 2 1 1862.4.a.e 2
8.b even 2 1 1216.4.a.g 2
8.d odd 2 1 1216.4.a.p 2
19.b odd 2 1 722.4.a.f 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - 9 T_{3} + 2$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(38))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - 2 T )^{2}$$
$3$ $$1 - 9 T + 56 T^{2} - 243 T^{3} + 729 T^{4}$$
$5$ $$1 + 9 T + 106 T^{2} + 1125 T^{3} + 15625 T^{4}$$
$7$ $$1 + 18 T + 475 T^{2} + 6174 T^{3} + 117649 T^{4}$$
$11$ $$1 + 17 T + 2716 T^{2} + 22627 T^{3} + 1771561 T^{4}$$
$13$ $$1 - 17 T + 1382 T^{2} - 37349 T^{3} + 4826809 T^{4}$$
$17$ $$1 + 80 T + 11353 T^{2} + 393040 T^{3} + 24137569 T^{4}$$
$19$ $$( 1 - 19 T )^{2}$$
$23$ $$1 - 73 T + 22582 T^{2} - 888191 T^{3} + 148035889 T^{4}$$
$29$ $$1 - 3 T + 40732 T^{2} - 73167 T^{3} + 594823321 T^{4}$$
$31$ $$1 - 212 T + 35486 T^{2} - 6315692 T^{3} + 887503681 T^{4}$$
$37$ $$1 - 192 T + 96214 T^{2} - 9725376 T^{3} + 2565726409 T^{4}$$
$41$ $$1 + 50 T + 136642 T^{2} + 3446050 T^{3} + 4750104241 T^{4}$$
$43$ $$1 - 677 T + 272702 T^{2} - 53826239 T^{3} + 6321363049 T^{4}$$
$47$ $$1 + 389 T + 153478 T^{2} + 40387147 T^{3} + 10779215329 T^{4}$$
$53$ $$1 + 1219 T + 663970 T^{2} + 181481063 T^{3} + 22164361129 T^{4}$$
$59$ $$1 + 287 T + 419944 T^{2} + 58943773 T^{3} + 42180533641 T^{4}$$
$61$ $$1 - 313 T + 253596 T^{2} - 71045053 T^{3} + 51520374361 T^{4}$$
$67$ $$1 - 1223 T + 867254 T^{2} - 367833149 T^{3} + 90458382169 T^{4}$$
$71$ $$1 - 200 T + 480250 T^{2} - 71582200 T^{3} + 128100283921 T^{4}$$
$73$ $$1 - 378 T + 195883 T^{2} - 147048426 T^{3} + 151334226289 T^{4}$$
$79$ $$1 - 1350 T + 1380310 T^{2} - 665602650 T^{3} + 243087455521 T^{4}$$
$83$ $$1 + 670 T + 568942 T^{2} + 383097290 T^{3} + 326940373369 T^{4}$$
$89$ $$1 + 236 T + 778834 T^{2} + 166372684 T^{3} + 496981290961 T^{4}$$
$97$ $$1 - 1294 T + 2054082 T^{2} - 1180998862 T^{3} + 832972004929 T^{4}$$