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

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: $$\Q(\sqrt{73})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 18$$ 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

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 $$\beta = \frac{1}{2}(1 + \sqrt{73})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 q^{2} + ( - \beta + 5) q^{3} + 4 q^{4} + (3 \beta - 6) q^{5} + ( - 2 \beta + 10) q^{6} + (4 \beta - 11) q^{7} + 8 q^{8} + ( - 9 \beta + 16) q^{9}+O(q^{10})$$ q + 2 * q^2 + (-b + 5) * q^3 + 4 * q^4 + (3*b - 6) * q^5 + (-2*b + 10) * q^6 + (4*b - 11) * q^7 + 8 * q^8 + (-9*b + 16) * q^9 $$q + 2 q^{2} + ( - \beta + 5) q^{3} + 4 q^{4} + (3 \beta - 6) q^{5} + ( - 2 \beta + 10) q^{6} + (4 \beta - 11) q^{7} + 8 q^{8} + ( - 9 \beta + 16) q^{9} + (6 \beta - 12) q^{10} + ( - \beta - 8) q^{11} + ( - 4 \beta + 20) q^{12} + ( - 13 \beta + 15) q^{13} + (8 \beta - 22) q^{14} + (18 \beta - 84) q^{15} + 16 q^{16} + (2 \beta - 41) q^{17} + ( - 18 \beta + 32) q^{18} + 19 q^{19} + (12 \beta - 24) q^{20} + (27 \beta - 127) q^{21} + ( - 2 \beta - 16) q^{22} + ( - 13 \beta + 43) q^{23} + ( - 8 \beta + 40) q^{24} + ( - 27 \beta + 73) q^{25} + ( - 26 \beta + 30) q^{26} + ( - 25 \beta + 107) q^{27} + (16 \beta - 44) q^{28} + (21 \beta - 9) q^{29} + (36 \beta - 168) q^{30} + (44 \beta + 84) q^{31} + 32 q^{32} + (4 \beta - 22) q^{33} + (4 \beta - 82) q^{34} + ( - 45 \beta + 282) q^{35} + ( - 36 \beta + 64) q^{36} + (28 \beta + 82) q^{37} + 38 q^{38} + ( - 67 \beta + 309) q^{39} + (24 \beta - 48) q^{40} + ( - 10 \beta - 20) q^{41} + (54 \beta - 254) q^{42} + ( - 7 \beta + 342) q^{43} + ( - 4 \beta - 32) q^{44} + (75 \beta - 582) q^{45} + ( - 26 \beta + 86) q^{46} + (71 \beta - 230) q^{47} + ( - 16 \beta + 80) q^{48} + ( - 72 \beta + 66) q^{49} + ( - 54 \beta + 146) q^{50} + (49 \beta - 241) q^{51} + ( - 52 \beta + 60) q^{52} + ( - 17 \beta - 601) q^{53} + ( - 50 \beta + 214) q^{54} + ( - 21 \beta - 6) q^{55} + (32 \beta - 88) q^{56} + ( - 19 \beta + 95) q^{57} + (42 \beta - 18) q^{58} + ( - 25 \beta - 131) q^{59} + (72 \beta - 336) q^{60} + ( - 111 \beta + 212) q^{61} + (88 \beta + 168) q^{62} + (127 \beta - 824) q^{63} + 64 q^{64} + (84 \beta - 792) q^{65} + (8 \beta - 44) q^{66} + (77 \beta + 573) q^{67} + (8 \beta - 164) q^{68} + ( - 95 \beta + 449) q^{69} + ( - 90 \beta + 564) q^{70} + ( - 116 \beta + 158) q^{71} + ( - 72 \beta + 128) q^{72} + (184 \beta + 97) q^{73} + (56 \beta + 164) q^{74} + ( - 181 \beta + 851) q^{75} + 76 q^{76} + ( - 25 \beta + 16) q^{77} + ( - 134 \beta + 618) q^{78} + (58 \beta + 646) q^{79} + (48 \beta - 96) q^{80} + (36 \beta + 553) q^{81} + ( - 20 \beta - 40) q^{82} + ( - 194 \beta - 238) q^{83} + (108 \beta - 508) q^{84} + ( - 129 \beta + 354) q^{85} + ( - 14 \beta + 684) q^{86} + (93 \beta - 423) q^{87} + ( - 8 \beta - 64) q^{88} + (188 \beta - 212) q^{89} + (150 \beta - 1164) q^{90} + (151 \beta - 1101) q^{91} + ( - 52 \beta + 172) q^{92} + (92 \beta - 372) q^{93} + (142 \beta - 460) q^{94} + (57 \beta - 114) q^{95} + ( - 32 \beta + 160) q^{96} + ( - 102 \beta + 698) q^{97} + ( - 144 \beta + 132) q^{98} + (65 \beta + 34) q^{99}+O(q^{100})$$ q + 2 * q^2 + (-b + 5) * q^3 + 4 * q^4 + (3*b - 6) * q^5 + (-2*b + 10) * q^6 + (4*b - 11) * q^7 + 8 * q^8 + (-9*b + 16) * q^9 + (6*b - 12) * q^10 + (-b - 8) * q^11 + (-4*b + 20) * q^12 + (-13*b + 15) * q^13 + (8*b - 22) * q^14 + (18*b - 84) * q^15 + 16 * q^16 + (2*b - 41) * q^17 + (-18*b + 32) * q^18 + 19 * q^19 + (12*b - 24) * q^20 + (27*b - 127) * q^21 + (-2*b - 16) * q^22 + (-13*b + 43) * q^23 + (-8*b + 40) * q^24 + (-27*b + 73) * q^25 + (-26*b + 30) * q^26 + (-25*b + 107) * q^27 + (16*b - 44) * q^28 + (21*b - 9) * q^29 + (36*b - 168) * q^30 + (44*b + 84) * q^31 + 32 * q^32 + (4*b - 22) * q^33 + (4*b - 82) * q^34 + (-45*b + 282) * q^35 + (-36*b + 64) * q^36 + (28*b + 82) * q^37 + 38 * q^38 + (-67*b + 309) * q^39 + (24*b - 48) * q^40 + (-10*b - 20) * q^41 + (54*b - 254) * q^42 + (-7*b + 342) * q^43 + (-4*b - 32) * q^44 + (75*b - 582) * q^45 + (-26*b + 86) * q^46 + (71*b - 230) * q^47 + (-16*b + 80) * q^48 + (-72*b + 66) * q^49 + (-54*b + 146) * q^50 + (49*b - 241) * q^51 + (-52*b + 60) * q^52 + (-17*b - 601) * q^53 + (-50*b + 214) * q^54 + (-21*b - 6) * q^55 + (32*b - 88) * q^56 + (-19*b + 95) * q^57 + (42*b - 18) * q^58 + (-25*b - 131) * q^59 + (72*b - 336) * q^60 + (-111*b + 212) * q^61 + (88*b + 168) * q^62 + (127*b - 824) * q^63 + 64 * q^64 + (84*b - 792) * q^65 + (8*b - 44) * q^66 + (77*b + 573) * q^67 + (8*b - 164) * q^68 + (-95*b + 449) * q^69 + (-90*b + 564) * q^70 + (-116*b + 158) * q^71 + (-72*b + 128) * q^72 + (184*b + 97) * q^73 + (56*b + 164) * q^74 + (-181*b + 851) * q^75 + 76 * q^76 + (-25*b + 16) * q^77 + (-134*b + 618) * q^78 + (58*b + 646) * q^79 + (48*b - 96) * q^80 + (36*b + 553) * q^81 + (-20*b - 40) * q^82 + (-194*b - 238) * q^83 + (108*b - 508) * q^84 + (-129*b + 354) * q^85 + (-14*b + 684) * q^86 + (93*b - 423) * q^87 + (-8*b - 64) * q^88 + (188*b - 212) * q^89 + (150*b - 1164) * q^90 + (151*b - 1101) * q^91 + (-52*b + 172) * q^92 + (92*b - 372) * q^93 + (142*b - 460) * q^94 + (57*b - 114) * q^95 + (-32*b + 160) * q^96 + (-102*b + 698) * q^97 + (-144*b + 132) * q^98 + (65*b + 34) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{2} + 9 q^{3} + 8 q^{4} - 9 q^{5} + 18 q^{6} - 18 q^{7} + 16 q^{8} + 23 q^{9}+O(q^{10})$$ 2 * q + 4 * q^2 + 9 * q^3 + 8 * q^4 - 9 * q^5 + 18 * q^6 - 18 * q^7 + 16 * q^8 + 23 * q^9 $$2 q + 4 q^{2} + 9 q^{3} + 8 q^{4} - 9 q^{5} + 18 q^{6} - 18 q^{7} + 16 q^{8} + 23 q^{9} - 18 q^{10} - 17 q^{11} + 36 q^{12} + 17 q^{13} - 36 q^{14} - 150 q^{15} + 32 q^{16} - 80 q^{17} + 46 q^{18} + 38 q^{19} - 36 q^{20} - 227 q^{21} - 34 q^{22} + 73 q^{23} + 72 q^{24} + 119 q^{25} + 34 q^{26} + 189 q^{27} - 72 q^{28} + 3 q^{29} - 300 q^{30} + 212 q^{31} + 64 q^{32} - 40 q^{33} - 160 q^{34} + 519 q^{35} + 92 q^{36} + 192 q^{37} + 76 q^{38} + 551 q^{39} - 72 q^{40} - 50 q^{41} - 454 q^{42} + 677 q^{43} - 68 q^{44} - 1089 q^{45} + 146 q^{46} - 389 q^{47} + 144 q^{48} + 60 q^{49} + 238 q^{50} - 433 q^{51} + 68 q^{52} - 1219 q^{53} + 378 q^{54} - 33 q^{55} - 144 q^{56} + 171 q^{57} + 6 q^{58} - 287 q^{59} - 600 q^{60} + 313 q^{61} + 424 q^{62} - 1521 q^{63} + 128 q^{64} - 1500 q^{65} - 80 q^{66} + 1223 q^{67} - 320 q^{68} + 803 q^{69} + 1038 q^{70} + 200 q^{71} + 184 q^{72} + 378 q^{73} + 384 q^{74} + 1521 q^{75} + 152 q^{76} + 7 q^{77} + 1102 q^{78} + 1350 q^{79} - 144 q^{80} + 1142 q^{81} - 100 q^{82} - 670 q^{83} - 908 q^{84} + 579 q^{85} + 1354 q^{86} - 753 q^{87} - 136 q^{88} - 236 q^{89} - 2178 q^{90} - 2051 q^{91} + 292 q^{92} - 652 q^{93} - 778 q^{94} - 171 q^{95} + 288 q^{96} + 1294 q^{97} + 120 q^{98} + 133 q^{99}+O(q^{100})$$ 2 * q + 4 * q^2 + 9 * q^3 + 8 * q^4 - 9 * q^5 + 18 * q^6 - 18 * q^7 + 16 * q^8 + 23 * q^9 - 18 * q^10 - 17 * q^11 + 36 * q^12 + 17 * q^13 - 36 * q^14 - 150 * q^15 + 32 * q^16 - 80 * q^17 + 46 * q^18 + 38 * q^19 - 36 * q^20 - 227 * q^21 - 34 * q^22 + 73 * q^23 + 72 * q^24 + 119 * q^25 + 34 * q^26 + 189 * q^27 - 72 * q^28 + 3 * q^29 - 300 * q^30 + 212 * q^31 + 64 * q^32 - 40 * q^33 - 160 * q^34 + 519 * q^35 + 92 * q^36 + 192 * q^37 + 76 * q^38 + 551 * q^39 - 72 * q^40 - 50 * q^41 - 454 * q^42 + 677 * q^43 - 68 * q^44 - 1089 * q^45 + 146 * q^46 - 389 * q^47 + 144 * q^48 + 60 * q^49 + 238 * q^50 - 433 * q^51 + 68 * q^52 - 1219 * q^53 + 378 * q^54 - 33 * q^55 - 144 * q^56 + 171 * q^57 + 6 * q^58 - 287 * q^59 - 600 * q^60 + 313 * q^61 + 424 * q^62 - 1521 * q^63 + 128 * q^64 - 1500 * q^65 - 80 * q^66 + 1223 * q^67 - 320 * q^68 + 803 * q^69 + 1038 * q^70 + 200 * q^71 + 184 * q^72 + 378 * q^73 + 384 * q^74 + 1521 * q^75 + 152 * q^76 + 7 * q^77 + 1102 * q^78 + 1350 * q^79 - 144 * q^80 + 1142 * q^81 - 100 * q^82 - 670 * q^83 - 908 * q^84 + 579 * q^85 + 1354 * q^86 - 753 * q^87 - 136 * q^88 - 236 * q^89 - 2178 * q^90 - 2051 * q^91 + 292 * q^92 - 652 * q^93 - 778 * q^94 - 171 * q^95 + 288 * q^96 + 1294 * q^97 + 120 * q^98 + 133 * 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
 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 $$T_{3}^{2} - 9T_{3} + 2$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(38))$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 2)^{2}$$
$3$ $$T^{2} - 9T + 2$$
$5$ $$T^{2} + 9T - 144$$
$7$ $$T^{2} + 18T - 211$$
$11$ $$T^{2} + 17T + 54$$
$13$ $$T^{2} - 17T - 3012$$
$17$ $$T^{2} + 80T + 1527$$
$19$ $$(T - 19)^{2}$$
$23$ $$T^{2} - 73T - 1752$$
$29$ $$T^{2} - 3T - 8046$$
$31$ $$T^{2} - 212T - 24096$$
$37$ $$T^{2} - 192T - 5092$$
$41$ $$T^{2} + 50T - 1200$$
$43$ $$T^{2} - 677T + 113688$$
$47$ $$T^{2} + 389T - 54168$$
$53$ $$T^{2} + 1219 T + 366216$$
$59$ $$T^{2} + 287T + 9186$$
$61$ $$T^{2} - 313T - 200366$$
$67$ $$T^{2} - 1223 T + 265728$$
$71$ $$T^{2} - 200T - 235572$$
$73$ $$T^{2} - 378T - 582151$$
$79$ $$T^{2} - 1350 T + 394232$$
$83$ $$T^{2} + 670T - 574632$$
$89$ $$T^{2} + 236T - 631104$$
$97$ $$T^{2} - 1294 T + 228736$$