# Properties

 Label 1862.4.a.e 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: $$\Q(\sqrt{73})$$ 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: 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 $$\beta = \frac{1}{2}(1 + \sqrt{73})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 q^{2} + ( - \beta - 4) q^{3} + 4 q^{4} + (3 \beta + 3) q^{5} + ( - 2 \beta - 8) q^{6} + 8 q^{8} + (9 \beta + 7) q^{9}+O(q^{10})$$ q + 2 * q^2 + (-b - 4) * q^3 + 4 * q^4 + (3*b + 3) * q^5 + (-2*b - 8) * q^6 + 8 * q^8 + (9*b + 7) * q^9 $$q + 2 q^{2} + ( - \beta - 4) q^{3} + 4 q^{4} + (3 \beta + 3) q^{5} + ( - 2 \beta - 8) q^{6} + 8 q^{8} + (9 \beta + 7) q^{9} + (6 \beta + 6) q^{10} + (\beta - 9) q^{11} + ( - 4 \beta - 16) q^{12} + ( - 13 \beta - 2) q^{13} + ( - 18 \beta - 66) q^{15} + 16 q^{16} + (2 \beta + 39) q^{17} + (18 \beta + 14) q^{18} - 19 q^{19} + (12 \beta + 12) q^{20} + (2 \beta - 18) q^{22} + (13 \beta + 30) q^{23} + ( - 8 \beta - 32) q^{24} + (27 \beta + 46) q^{25} + ( - 26 \beta - 4) q^{26} + ( - 25 \beta - 82) q^{27} + ( - 21 \beta + 12) q^{29} + ( - 36 \beta - 132) q^{30} + (44 \beta - 128) q^{31} + 32 q^{32} + (4 \beta + 18) q^{33} + (4 \beta + 78) q^{34} + (36 \beta + 28) q^{36} + ( - 28 \beta + 110) q^{37} - 38 q^{38} + (67 \beta + 242) q^{39} + (24 \beta + 24) q^{40} + ( - 10 \beta + 30) q^{41} + (7 \beta + 335) q^{43} + (4 \beta - 36) q^{44} + (75 \beta + 507) q^{45} + (26 \beta + 60) q^{46} + (71 \beta + 159) q^{47} + ( - 16 \beta - 64) q^{48} + (54 \beta + 92) q^{50} + ( - 49 \beta - 192) q^{51} + ( - 52 \beta - 8) q^{52} + (17 \beta - 618) q^{53} + ( - 50 \beta - 164) q^{54} + ( - 21 \beta + 27) q^{55} + (19 \beta + 76) q^{57} + ( - 42 \beta + 24) q^{58} + ( - 25 \beta + 156) q^{59} + ( - 72 \beta - 264) q^{60} + ( - 111 \beta - 101) q^{61} + (88 \beta - 256) q^{62} + 64 q^{64} + ( - 84 \beta - 708) q^{65} + (8 \beta + 36) q^{66} + ( - 77 \beta + 650) q^{67} + (8 \beta + 156) q^{68} + ( - 95 \beta - 354) q^{69} + (116 \beta + 42) q^{71} + (72 \beta + 56) q^{72} + (184 \beta - 281) q^{73} + ( - 56 \beta + 220) q^{74} + ( - 181 \beta - 670) q^{75} - 76 q^{76} + (134 \beta + 484) q^{78} + ( - 58 \beta + 704) q^{79} + (48 \beta + 48) q^{80} + ( - 36 \beta + 589) q^{81} + ( - 20 \beta + 60) q^{82} + ( - 194 \beta + 432) q^{83} + (129 \beta + 225) q^{85} + (14 \beta + 670) q^{86} + (93 \beta + 330) q^{87} + (8 \beta - 72) q^{88} + (188 \beta + 24) q^{89} + (150 \beta + 1014) q^{90} + (52 \beta + 120) q^{92} + ( - 92 \beta - 280) q^{93} + (142 \beta + 318) q^{94} + ( - 57 \beta - 57) q^{95} + ( - 32 \beta - 128) q^{96} + ( - 102 \beta - 596) q^{97} + ( - 65 \beta + 99) q^{99}+O(q^{100})$$ q + 2 * q^2 + (-b - 4) * q^3 + 4 * q^4 + (3*b + 3) * q^5 + (-2*b - 8) * q^6 + 8 * q^8 + (9*b + 7) * q^9 + (6*b + 6) * q^10 + (b - 9) * q^11 + (-4*b - 16) * q^12 + (-13*b - 2) * q^13 + (-18*b - 66) * q^15 + 16 * q^16 + (2*b + 39) * q^17 + (18*b + 14) * q^18 - 19 * q^19 + (12*b + 12) * q^20 + (2*b - 18) * q^22 + (13*b + 30) * q^23 + (-8*b - 32) * q^24 + (27*b + 46) * q^25 + (-26*b - 4) * q^26 + (-25*b - 82) * q^27 + (-21*b + 12) * q^29 + (-36*b - 132) * q^30 + (44*b - 128) * q^31 + 32 * q^32 + (4*b + 18) * q^33 + (4*b + 78) * q^34 + (36*b + 28) * q^36 + (-28*b + 110) * q^37 - 38 * q^38 + (67*b + 242) * q^39 + (24*b + 24) * q^40 + (-10*b + 30) * q^41 + (7*b + 335) * q^43 + (4*b - 36) * q^44 + (75*b + 507) * q^45 + (26*b + 60) * q^46 + (71*b + 159) * q^47 + (-16*b - 64) * q^48 + (54*b + 92) * q^50 + (-49*b - 192) * q^51 + (-52*b - 8) * q^52 + (17*b - 618) * q^53 + (-50*b - 164) * q^54 + (-21*b + 27) * q^55 + (19*b + 76) * q^57 + (-42*b + 24) * q^58 + (-25*b + 156) * q^59 + (-72*b - 264) * q^60 + (-111*b - 101) * q^61 + (88*b - 256) * q^62 + 64 * q^64 + (-84*b - 708) * q^65 + (8*b + 36) * q^66 + (-77*b + 650) * q^67 + (8*b + 156) * q^68 + (-95*b - 354) * q^69 + (116*b + 42) * q^71 + (72*b + 56) * q^72 + (184*b - 281) * q^73 + (-56*b + 220) * q^74 + (-181*b - 670) * q^75 - 76 * q^76 + (134*b + 484) * q^78 + (-58*b + 704) * q^79 + (48*b + 48) * q^80 + (-36*b + 589) * q^81 + (-20*b + 60) * q^82 + (-194*b + 432) * q^83 + (129*b + 225) * q^85 + (14*b + 670) * q^86 + (93*b + 330) * q^87 + (8*b - 72) * q^88 + (188*b + 24) * q^89 + (150*b + 1014) * q^90 + (52*b + 120) * q^92 + (-92*b - 280) * q^93 + (142*b + 318) * q^94 + (-57*b - 57) * q^95 + (-32*b - 128) * q^96 + (-102*b - 596) * q^97 + (-65*b + 99) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{2} - 9 q^{3} + 8 q^{4} + 9 q^{5} - 18 q^{6} + 16 q^{8} + 23 q^{9}+O(q^{10})$$ 2 * q + 4 * q^2 - 9 * q^3 + 8 * q^4 + 9 * q^5 - 18 * q^6 + 16 * q^8 + 23 * q^9 $$2 q + 4 q^{2} - 9 q^{3} + 8 q^{4} + 9 q^{5} - 18 q^{6} + 16 q^{8} + 23 q^{9} + 18 q^{10} - 17 q^{11} - 36 q^{12} - 17 q^{13} - 150 q^{15} + 32 q^{16} + 80 q^{17} + 46 q^{18} - 38 q^{19} + 36 q^{20} - 34 q^{22} + 73 q^{23} - 72 q^{24} + 119 q^{25} - 34 q^{26} - 189 q^{27} + 3 q^{29} - 300 q^{30} - 212 q^{31} + 64 q^{32} + 40 q^{33} + 160 q^{34} + 92 q^{36} + 192 q^{37} - 76 q^{38} + 551 q^{39} + 72 q^{40} + 50 q^{41} + 677 q^{43} - 68 q^{44} + 1089 q^{45} + 146 q^{46} + 389 q^{47} - 144 q^{48} + 238 q^{50} - 433 q^{51} - 68 q^{52} - 1219 q^{53} - 378 q^{54} + 33 q^{55} + 171 q^{57} + 6 q^{58} + 287 q^{59} - 600 q^{60} - 313 q^{61} - 424 q^{62} + 128 q^{64} - 1500 q^{65} + 80 q^{66} + 1223 q^{67} + 320 q^{68} - 803 q^{69} + 200 q^{71} + 184 q^{72} - 378 q^{73} + 384 q^{74} - 1521 q^{75} - 152 q^{76} + 1102 q^{78} + 1350 q^{79} + 144 q^{80} + 1142 q^{81} + 100 q^{82} + 670 q^{83} + 579 q^{85} + 1354 q^{86} + 753 q^{87} - 136 q^{88} + 236 q^{89} + 2178 q^{90} + 292 q^{92} - 652 q^{93} + 778 q^{94} - 171 q^{95} - 288 q^{96} - 1294 q^{97} + 133 q^{99}+O(q^{100})$$ 2 * q + 4 * q^2 - 9 * q^3 + 8 * q^4 + 9 * q^5 - 18 * q^6 + 16 * q^8 + 23 * q^9 + 18 * q^10 - 17 * q^11 - 36 * q^12 - 17 * q^13 - 150 * q^15 + 32 * q^16 + 80 * q^17 + 46 * q^18 - 38 * q^19 + 36 * q^20 - 34 * q^22 + 73 * q^23 - 72 * q^24 + 119 * q^25 - 34 * q^26 - 189 * q^27 + 3 * q^29 - 300 * q^30 - 212 * q^31 + 64 * q^32 + 40 * q^33 + 160 * q^34 + 92 * q^36 + 192 * q^37 - 76 * q^38 + 551 * q^39 + 72 * q^40 + 50 * q^41 + 677 * q^43 - 68 * q^44 + 1089 * q^45 + 146 * q^46 + 389 * q^47 - 144 * q^48 + 238 * q^50 - 433 * q^51 - 68 * q^52 - 1219 * q^53 - 378 * q^54 + 33 * q^55 + 171 * q^57 + 6 * q^58 + 287 * q^59 - 600 * q^60 - 313 * q^61 - 424 * q^62 + 128 * q^64 - 1500 * q^65 + 80 * q^66 + 1223 * q^67 + 320 * q^68 - 803 * q^69 + 200 * q^71 + 184 * q^72 - 378 * q^73 + 384 * q^74 - 1521 * q^75 - 152 * q^76 + 1102 * q^78 + 1350 * q^79 + 144 * q^80 + 1142 * q^81 + 100 * q^82 + 670 * q^83 + 579 * q^85 + 1354 * q^86 + 753 * q^87 - 136 * q^88 + 236 * q^89 + 2178 * q^90 + 292 * q^92 - 652 * q^93 + 778 * q^94 - 171 * q^95 - 288 * q^96 - 1294 * q^97 + 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 −8.77200 4.00000 17.3160 −17.5440 0 8.00000 49.9480 34.6320
1.2 2.00000 −0.227998 4.00000 −8.31601 −0.455996 0 8.00000 −26.9480 −16.6320
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

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

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

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 9T_{3} + 2$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1862))$$.

## Hecke characteristic polynomials

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