# Properties

 Label 342.4.a.k Level $342$ Weight $4$ Character orbit 342.a Self dual yes Analytic conductor $20.179$ 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] = [342,4,Mod(1,342)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(342, 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("342.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$342 = 2 \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 342.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: $$20.1786532220$$ 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: $$\Z[a_1, \ldots, a_{7}]$$ 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.

 $$f(q)$$ $$=$$ $$q + 2 q^{2} + 4 q^{4} + ( - 2 \beta - 4) q^{5} + ( - \beta + 29) q^{7} + 8 q^{8}+O(q^{10})$$ q + 2 * q^2 + 4 * q^4 + (-2*b - 4) * q^5 + (-b + 29) * q^7 + 8 * q^8 $$q + 2 q^{2} + 4 q^{4} + ( - 2 \beta - 4) q^{5} + ( - \beta + 29) q^{7} + 8 q^{8} + ( - 4 \beta - 8) q^{10} + (2 \beta - 6) q^{11} + (7 \beta + 3) q^{13} + ( - 2 \beta + 58) q^{14} + 16 q^{16} + ( - 15 \beta + 33) q^{17} - 19 q^{19} + ( - 8 \beta - 16) q^{20} + (4 \beta - 12) q^{22} + (13 \beta + 71) q^{23} + (20 \beta + 67) q^{25} + (14 \beta + 6) q^{26} + ( - 4 \beta + 116) q^{28} + (29 \beta + 25) q^{29} + (16 \beta - 16) q^{31} + 32 q^{32} + ( - 30 \beta + 66) q^{34} + ( - 52 \beta - 28) q^{35} + (16 \beta + 182) q^{37} - 38 q^{38} + ( - 16 \beta - 32) q^{40} + (6 \beta + 392) q^{41} + ( - 48 \beta + 172) q^{43} + (8 \beta - 24) q^{44} + (26 \beta + 142) q^{46} + (40 \beta + 80) q^{47} + ( - 57 \beta + 542) q^{49} + (40 \beta + 134) q^{50} + (28 \beta + 12) q^{52} + (9 \beta - 203) q^{53} - 152 q^{55} + ( - 8 \beta + 232) q^{56} + (58 \beta + 50) q^{58} + ( - 71 \beta - 65) q^{59} + ( - 44 \beta - 318) q^{61} + (32 \beta - 32) q^{62} + 64 q^{64} + ( - 48 \beta - 628) q^{65} + (43 \beta - 491) q^{67} + ( - 60 \beta + 132) q^{68} + ( - 104 \beta - 56) q^{70} + (22 \beta - 214) q^{71} + (\beta + 61) q^{73} + (32 \beta + 364) q^{74} - 76 q^{76} + (62 \beta - 262) q^{77} + ( - 58 \beta + 82) q^{79} + ( - 32 \beta - 64) q^{80} + (12 \beta + 784) q^{82} + ( - 6 \beta - 1110) q^{83} + (24 \beta + 1188) q^{85} + ( - 96 \beta + 344) q^{86} + (16 \beta - 48) q^{88} + ( - 10 \beta + 440) q^{89} + (193 \beta - 221) q^{91} + (52 \beta + 284) q^{92} + (80 \beta + 160) q^{94} + (38 \beta + 76) q^{95} + (76 \beta - 970) q^{97} + ( - 114 \beta + 1084) q^{98}+O(q^{100})$$ q + 2 * q^2 + 4 * q^4 + (-2*b - 4) * q^5 + (-b + 29) * q^7 + 8 * q^8 + (-4*b - 8) * q^10 + (2*b - 6) * q^11 + (7*b + 3) * q^13 + (-2*b + 58) * q^14 + 16 * q^16 + (-15*b + 33) * q^17 - 19 * q^19 + (-8*b - 16) * q^20 + (4*b - 12) * q^22 + (13*b + 71) * q^23 + (20*b + 67) * q^25 + (14*b + 6) * q^26 + (-4*b + 116) * q^28 + (29*b + 25) * q^29 + (16*b - 16) * q^31 + 32 * q^32 + (-30*b + 66) * q^34 + (-52*b - 28) * q^35 + (16*b + 182) * q^37 - 38 * q^38 + (-16*b - 32) * q^40 + (6*b + 392) * q^41 + (-48*b + 172) * q^43 + (8*b - 24) * q^44 + (26*b + 142) * q^46 + (40*b + 80) * q^47 + (-57*b + 542) * q^49 + (40*b + 134) * q^50 + (28*b + 12) * q^52 + (9*b - 203) * q^53 - 152 * q^55 + (-8*b + 232) * q^56 + (58*b + 50) * q^58 + (-71*b - 65) * q^59 + (-44*b - 318) * q^61 + (32*b - 32) * q^62 + 64 * q^64 + (-48*b - 628) * q^65 + (43*b - 491) * q^67 + (-60*b + 132) * q^68 + (-104*b - 56) * q^70 + (22*b - 214) * q^71 + (b + 61) * q^73 + (32*b + 364) * q^74 - 76 * q^76 + (62*b - 262) * q^77 + (-58*b + 82) * q^79 + (-32*b - 64) * q^80 + (12*b + 784) * q^82 + (-6*b - 1110) * q^83 + (24*b + 1188) * q^85 + (-96*b + 344) * q^86 + (16*b - 48) * q^88 + (-10*b + 440) * q^89 + (193*b - 221) * q^91 + (52*b + 284) * q^92 + (80*b + 160) * q^94 + (38*b + 76) * q^95 + (76*b - 970) * q^97 + (-114*b + 1084) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{2} + 8 q^{4} - 10 q^{5} + 57 q^{7} + 16 q^{8}+O(q^{10})$$ 2 * q + 4 * q^2 + 8 * q^4 - 10 * q^5 + 57 * q^7 + 16 * q^8 $$2 q + 4 q^{2} + 8 q^{4} - 10 q^{5} + 57 q^{7} + 16 q^{8} - 20 q^{10} - 10 q^{11} + 13 q^{13} + 114 q^{14} + 32 q^{16} + 51 q^{17} - 38 q^{19} - 40 q^{20} - 20 q^{22} + 155 q^{23} + 154 q^{25} + 26 q^{26} + 228 q^{28} + 79 q^{29} - 16 q^{31} + 64 q^{32} + 102 q^{34} - 108 q^{35} + 380 q^{37} - 76 q^{38} - 80 q^{40} + 790 q^{41} + 296 q^{43} - 40 q^{44} + 310 q^{46} + 200 q^{47} + 1027 q^{49} + 308 q^{50} + 52 q^{52} - 397 q^{53} - 304 q^{55} + 456 q^{56} + 158 q^{58} - 201 q^{59} - 680 q^{61} - 32 q^{62} + 128 q^{64} - 1304 q^{65} - 939 q^{67} + 204 q^{68} - 216 q^{70} - 406 q^{71} + 123 q^{73} + 760 q^{74} - 152 q^{76} - 462 q^{77} + 106 q^{79} - 160 q^{80} + 1580 q^{82} - 2226 q^{83} + 2400 q^{85} + 592 q^{86} - 80 q^{88} + 870 q^{89} - 249 q^{91} + 620 q^{92} + 400 q^{94} + 190 q^{95} - 1864 q^{97} + 2054 q^{98}+O(q^{100})$$ 2 * q + 4 * q^2 + 8 * q^4 - 10 * q^5 + 57 * q^7 + 16 * q^8 - 20 * q^10 - 10 * q^11 + 13 * q^13 + 114 * q^14 + 32 * q^16 + 51 * q^17 - 38 * q^19 - 40 * q^20 - 20 * q^22 + 155 * q^23 + 154 * q^25 + 26 * q^26 + 228 * q^28 + 79 * q^29 - 16 * q^31 + 64 * q^32 + 102 * q^34 - 108 * q^35 + 380 * q^37 - 76 * q^38 - 80 * q^40 + 790 * q^41 + 296 * q^43 - 40 * q^44 + 310 * q^46 + 200 * q^47 + 1027 * q^49 + 308 * q^50 + 52 * q^52 - 397 * q^53 - 304 * q^55 + 456 * q^56 + 158 * q^58 - 201 * q^59 - 680 * q^61 - 32 * q^62 + 128 * q^64 - 1304 * q^65 - 939 * q^67 + 204 * q^68 - 216 * q^70 - 406 * q^71 + 123 * q^73 + 760 * q^74 - 152 * q^76 - 462 * q^77 + 106 * q^79 - 160 * q^80 + 1580 * q^82 - 2226 * q^83 + 2400 * q^85 + 592 * q^86 - 80 * q^88 + 870 * q^89 - 249 * q^91 + 620 * q^92 + 400 * q^94 + 190 * q^95 - 1864 * q^97 + 2054 * q^98

## 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 0 4.00000 −18.3041 0 21.8479 8.00000 0 −36.6083
1.2 2.00000 0 4.00000 8.30413 0 35.1521 8.00000 0 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$19$$ $$1$$

## 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 342.4.a.k 2
3.b odd 2 1 38.4.a.b 2
12.b even 2 1 304.4.a.d 2
15.d odd 2 1 950.4.a.h 2
15.e even 4 2 950.4.b.g 4
21.c even 2 1 1862.4.a.b 2
24.f even 2 1 1216.4.a.l 2
24.h odd 2 1 1216.4.a.j 2
57.d 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 3.b odd 2 1
304.4.a.d 2 12.b even 2 1
342.4.a.k 2 1.a even 1 1 trivial
722.4.a.i 2 57.d even 2 1
950.4.a.h 2 15.d odd 2 1
950.4.b.g 4 15.e even 4 2
1216.4.a.j 2 24.h odd 2 1
1216.4.a.l 2 24.f even 2 1
1862.4.a.b 2 21.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 10T_{5} - 152$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(342))$$.

## Hecke characteristic polynomials

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