# 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

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [38,4,Mod(1,38)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(38, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("38.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

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

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$2.24207258022$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{177})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 44$$ 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

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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.

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

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

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

$p$ $F_p(T)$
$2$ $$(T + 2)^{2}$$
$3$ $$T^{2} - T - 44$$
$5$ $$T^{2} - 10T - 152$$
$7$ $$T^{2} - 57T + 768$$
$11$ $$T^{2} - 10T - 152$$
$13$ $$T^{2} - 13T - 2126$$
$17$ $$T^{2} + 51T - 9306$$
$19$ $$(T + 19)^{2}$$
$23$ $$T^{2} + 155T - 1472$$
$29$ $$T^{2} + 79T - 35654$$
$31$ $$T^{2} + 16T - 11264$$
$37$ $$T^{2} - 380T + 24772$$
$41$ $$T^{2} + 790T + 154432$$
$43$ $$T^{2} - 296T - 80048$$
$47$ $$T^{2} + 200T - 60800$$
$53$ $$T^{2} - 397T + 35818$$
$59$ $$T^{2} - 201T - 212964$$
$61$ $$T^{2} + 680T + 29932$$
$67$ $$T^{2} + 939T + 138612$$
$71$ $$T^{2} - 406T + 19792$$
$73$ $$T^{2} - 123T + 3738$$
$79$ $$T^{2} - 106T - 146048$$
$83$ $$T^{2} - 2226 T + 1237176$$
$89$ $$T^{2} + 870T + 184800$$
$97$ $$T^{2} + 1864 T + 613036$$