# Properties

 Label 1862.4.a.b Level $1862$ Weight $4$ Character orbit 1862.a Self dual yes Analytic conductor $109.862$ 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] = [1862,4,Mod(1,1862)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1862, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1862.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1862 = 2 \cdot 7^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1862.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: $$109.861556431$$ 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: no (minimal twist has level 38) 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.

 $$f(q)$$ $$=$$ $$q - 2 q^{2} - \beta q^{3} + 4 q^{4} + (2 \beta - 6) q^{5} + 2 \beta q^{6} - 8 q^{8} + (\beta + 17) q^{9} +O(q^{10})$$ q - 2 * q^2 - b * q^3 + 4 * q^4 + (2*b - 6) * q^5 + 2*b * q^6 - 8 * q^8 + (b + 17) * q^9 $$q - 2 q^{2} - \beta q^{3} + 4 q^{4} + (2 \beta - 6) q^{5} + 2 \beta q^{6} - 8 q^{8} + (\beta + 17) q^{9} + ( - 4 \beta + 12) q^{10} + (2 \beta + 4) q^{11} - 4 \beta q^{12} + (7 \beta - 10) q^{13} + (4 \beta - 88) q^{15} + 16 q^{16} + (15 \beta + 18) q^{17} + ( - 2 \beta - 34) q^{18} + 19 q^{19} + (8 \beta - 24) q^{20} + ( - 4 \beta - 8) q^{22} + (13 \beta - 84) q^{23} + 8 \beta q^{24} + ( - 20 \beta + 87) q^{25} + ( - 14 \beta + 20) q^{26} + (9 \beta - 44) q^{27} + (29 \beta - 54) q^{29} + ( - 8 \beta + 176) q^{30} + 16 \beta q^{31} - 32 q^{32} + ( - 6 \beta - 88) q^{33} + ( - 30 \beta - 36) q^{34} + (4 \beta + 68) q^{36} + ( - 16 \beta + 198) q^{37} - 38 q^{38} + (3 \beta - 308) q^{39} + ( - 16 \beta + 48) q^{40} + ( - 6 \beta + 398) q^{41} + (48 \beta + 124) q^{43} + (8 \beta + 16) q^{44} + (30 \beta - 14) q^{45} + ( - 26 \beta + 168) q^{46} + ( - 40 \beta + 120) q^{47} - 16 \beta q^{48} + (40 \beta - 174) q^{50} + ( - 33 \beta - 660) q^{51} + (28 \beta - 40) q^{52} + (9 \beta + 194) q^{53} + ( - 18 \beta + 88) q^{54} + 152 q^{55} - 19 \beta q^{57} + ( - 58 \beta + 108) q^{58} + (71 \beta - 136) q^{59} + (16 \beta - 352) q^{60} + ( - 44 \beta + 362) q^{61} - 32 \beta q^{62} + 64 q^{64} + ( - 48 \beta + 676) q^{65} + (12 \beta + 176) q^{66} + ( - 43 \beta - 448) q^{67} + (60 \beta + 72) q^{68} + (71 \beta - 572) q^{69} + (22 \beta + 192) q^{71} + ( - 8 \beta - 136) q^{72} + (\beta - 62) q^{73} + (32 \beta - 396) q^{74} + ( - 67 \beta + 880) q^{75} + 76 q^{76} + ( - 6 \beta + 616) q^{78} + (58 \beta + 24) q^{79} + (32 \beta - 96) q^{80} + (8 \beta - 855) q^{81} + (12 \beta - 796) q^{82} + (6 \beta - 1116) q^{83} + ( - 24 \beta + 1212) q^{85} + ( - 96 \beta - 248) q^{86} + (25 \beta - 1276) q^{87} + ( - 16 \beta - 32) q^{88} + (10 \beta + 430) q^{89} + ( - 60 \beta + 28) q^{90} + (52 \beta - 336) q^{92} + ( - 16 \beta - 704) q^{93} + (80 \beta - 240) q^{94} + (38 \beta - 114) q^{95} + 32 \beta q^{96} + (76 \beta + 894) q^{97} + (40 \beta + 156) q^{99} +O(q^{100})$$ q - 2 * q^2 - b * q^3 + 4 * q^4 + (2*b - 6) * q^5 + 2*b * q^6 - 8 * q^8 + (b + 17) * q^9 + (-4*b + 12) * q^10 + (2*b + 4) * q^11 - 4*b * q^12 + (7*b - 10) * q^13 + (4*b - 88) * q^15 + 16 * q^16 + (15*b + 18) * q^17 + (-2*b - 34) * q^18 + 19 * q^19 + (8*b - 24) * q^20 + (-4*b - 8) * q^22 + (13*b - 84) * q^23 + 8*b * q^24 + (-20*b + 87) * q^25 + (-14*b + 20) * q^26 + (9*b - 44) * q^27 + (29*b - 54) * q^29 + (-8*b + 176) * q^30 + 16*b * q^31 - 32 * q^32 + (-6*b - 88) * q^33 + (-30*b - 36) * q^34 + (4*b + 68) * q^36 + (-16*b + 198) * q^37 - 38 * q^38 + (3*b - 308) * q^39 + (-16*b + 48) * q^40 + (-6*b + 398) * q^41 + (48*b + 124) * q^43 + (8*b + 16) * q^44 + (30*b - 14) * q^45 + (-26*b + 168) * q^46 + (-40*b + 120) * q^47 - 16*b * q^48 + (40*b - 174) * q^50 + (-33*b - 660) * q^51 + (28*b - 40) * q^52 + (9*b + 194) * q^53 + (-18*b + 88) * q^54 + 152 * q^55 - 19*b * q^57 + (-58*b + 108) * q^58 + (71*b - 136) * q^59 + (16*b - 352) * q^60 + (-44*b + 362) * q^61 - 32*b * q^62 + 64 * q^64 + (-48*b + 676) * q^65 + (12*b + 176) * q^66 + (-43*b - 448) * q^67 + (60*b + 72) * q^68 + (71*b - 572) * q^69 + (22*b + 192) * q^71 + (-8*b - 136) * q^72 + (b - 62) * q^73 + (32*b - 396) * q^74 + (-67*b + 880) * q^75 + 76 * q^76 + (-6*b + 616) * q^78 + (58*b + 24) * q^79 + (32*b - 96) * q^80 + (8*b - 855) * q^81 + (12*b - 796) * q^82 + (6*b - 1116) * q^83 + (-24*b + 1212) * q^85 + (-96*b - 248) * q^86 + (25*b - 1276) * q^87 + (-16*b - 32) * q^88 + (10*b + 430) * q^89 + (-60*b + 28) * q^90 + (52*b - 336) * q^92 + (-16*b - 704) * q^93 + (80*b - 240) * q^94 + (38*b - 114) * q^95 + 32*b * q^96 + (76*b + 894) * q^97 + (40*b + 156) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{2} - q^{3} + 8 q^{4} - 10 q^{5} + 2 q^{6} - 16 q^{8} + 35 q^{9}+O(q^{10})$$ 2 * q - 4 * q^2 - q^3 + 8 * q^4 - 10 * q^5 + 2 * q^6 - 16 * q^8 + 35 * q^9 $$2 q - 4 q^{2} - q^{3} + 8 q^{4} - 10 q^{5} + 2 q^{6} - 16 q^{8} + 35 q^{9} + 20 q^{10} + 10 q^{11} - 4 q^{12} - 13 q^{13} - 172 q^{15} + 32 q^{16} + 51 q^{17} - 70 q^{18} + 38 q^{19} - 40 q^{20} - 20 q^{22} - 155 q^{23} + 8 q^{24} + 154 q^{25} + 26 q^{26} - 79 q^{27} - 79 q^{29} + 344 q^{30} + 16 q^{31} - 64 q^{32} - 182 q^{33} - 102 q^{34} + 140 q^{36} + 380 q^{37} - 76 q^{38} - 613 q^{39} + 80 q^{40} + 790 q^{41} + 296 q^{43} + 40 q^{44} + 2 q^{45} + 310 q^{46} + 200 q^{47} - 16 q^{48} - 308 q^{50} - 1353 q^{51} - 52 q^{52} + 397 q^{53} + 158 q^{54} + 304 q^{55} - 19 q^{57} + 158 q^{58} - 201 q^{59} - 688 q^{60} + 680 q^{61} - 32 q^{62} + 128 q^{64} + 1304 q^{65} + 364 q^{66} - 939 q^{67} + 204 q^{68} - 1073 q^{69} + 406 q^{71} - 280 q^{72} - 123 q^{73} - 760 q^{74} + 1693 q^{75} + 152 q^{76} + 1226 q^{78} + 106 q^{79} - 160 q^{80} - 1702 q^{81} - 1580 q^{82} - 2226 q^{83} + 2400 q^{85} - 592 q^{86} - 2527 q^{87} - 80 q^{88} + 870 q^{89} - 4 q^{90} - 620 q^{92} - 1424 q^{93} - 400 q^{94} - 190 q^{95} + 32 q^{96} + 1864 q^{97} + 352 q^{99}+O(q^{100})$$ 2 * q - 4 * q^2 - q^3 + 8 * q^4 - 10 * q^5 + 2 * q^6 - 16 * q^8 + 35 * q^9 + 20 * q^10 + 10 * q^11 - 4 * q^12 - 13 * q^13 - 172 * q^15 + 32 * q^16 + 51 * q^17 - 70 * q^18 + 38 * q^19 - 40 * q^20 - 20 * q^22 - 155 * q^23 + 8 * q^24 + 154 * q^25 + 26 * q^26 - 79 * q^27 - 79 * q^29 + 344 * q^30 + 16 * q^31 - 64 * q^32 - 182 * q^33 - 102 * q^34 + 140 * q^36 + 380 * q^37 - 76 * q^38 - 613 * q^39 + 80 * q^40 + 790 * q^41 + 296 * q^43 + 40 * q^44 + 2 * q^45 + 310 * q^46 + 200 * q^47 - 16 * q^48 - 308 * q^50 - 1353 * q^51 - 52 * q^52 + 397 * q^53 + 158 * q^54 + 304 * q^55 - 19 * q^57 + 158 * q^58 - 201 * q^59 - 688 * q^60 + 680 * q^61 - 32 * q^62 + 128 * q^64 + 1304 * q^65 + 364 * q^66 - 939 * q^67 + 204 * q^68 - 1073 * q^69 + 406 * q^71 - 280 * q^72 - 123 * q^73 - 760 * q^74 + 1693 * q^75 + 152 * q^76 + 1226 * q^78 + 106 * q^79 - 160 * q^80 - 1702 * q^81 - 1580 * q^82 - 2226 * q^83 + 2400 * q^85 - 592 * q^86 - 2527 * q^87 - 80 * q^88 + 870 * q^89 - 4 * q^90 - 620 * q^92 - 1424 * q^93 - 400 * q^94 - 190 * q^95 + 32 * q^96 + 1864 * q^97 + 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 −7.15207 4.00000 8.30413 14.3041 0 −8.00000 24.1521 −16.6083
1.2 −2.00000 6.15207 4.00000 −18.3041 −12.3041 0 −8.00000 10.8479 36.6083
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$7$$ $$-1$$
$$19$$ $$-1$$

## 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 1862.4.a.b 2
7.b odd 2 1 38.4.a.b 2
21.c even 2 1 342.4.a.k 2
28.d even 2 1 304.4.a.d 2
35.c odd 2 1 950.4.a.h 2
35.f even 4 2 950.4.b.g 4
56.e even 2 1 1216.4.a.l 2
56.h odd 2 1 1216.4.a.j 2
133.c even 2 1 722.4.a.i 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.4.a.b 2 7.b odd 2 1
304.4.a.d 2 28.d even 2 1
342.4.a.k 2 21.c even 2 1
722.4.a.i 2 133.c even 2 1
950.4.a.h 2 35.c odd 2 1
950.4.b.g 4 35.f even 4 2
1216.4.a.j 2 56.h odd 2 1
1216.4.a.l 2 56.e even 2 1
1862.4.a.b 2 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + T_{3} - 44$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1862))$$.

## Hecke characteristic polynomials

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