# Properties

 Label 1344.4.a.bo Level $1344$ Weight $4$ Character orbit 1344.a Self dual yes Analytic conductor $79.299$ 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] = [1344,4,Mod(1,1344)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1344, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("1344.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1344 = 2^{6} \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1344.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: $$79.2985670477$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{57})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 14$$ x^2 - x - 14 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 21) 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 $$\beta = \sqrt{57}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 3 q^{3} + ( - \beta - 3) q^{5} - 7 q^{7} + 9 q^{9}+O(q^{10})$$ q + 3 * q^3 + (-b - 3) * q^5 - 7 * q^7 + 9 * q^9 $$q + 3 q^{3} + ( - \beta - 3) q^{5} - 7 q^{7} + 9 q^{9} + (5 \beta - 3) q^{11} + (6 \beta - 8) q^{13} + ( - 3 \beta - 9) q^{15} + ( - \beta - 3) q^{17} + ( - 12 \beta + 32) q^{19} - 21 q^{21} + (17 \beta - 3) q^{23} + (6 \beta - 59) q^{25} + 27 q^{27} + ( - 12 \beta + 126) q^{29} + ( - 36 \beta - 20) q^{31} + (15 \beta - 9) q^{33} + (7 \beta + 21) q^{35} + (18 \beta + 124) q^{37} + (18 \beta - 24) q^{39} + ( - 15 \beta - 225) q^{41} + ( - 24 \beta + 188) q^{43} + ( - 9 \beta - 27) q^{45} + ( - 34 \beta + 6) q^{47} + 49 q^{49} + ( - 3 \beta - 9) q^{51} + ( - 2 \beta + 552) q^{53} + ( - 12 \beta - 276) q^{55} + ( - 36 \beta + 96) q^{57} + ( - 58 \beta + 402) q^{59} + ( - 36 \beta + 214) q^{61} - 63 q^{63} + ( - 10 \beta - 318) q^{65} + (54 \beta + 74) q^{67} + (51 \beta - 9) q^{69} + (15 \beta - 477) q^{71} + (6 \beta + 536) q^{73} + (18 \beta - 177) q^{75} + ( - 35 \beta + 21) q^{77} + (54 \beta + 286) q^{79} + 81 q^{81} + (48 \beta + 972) q^{83} + (6 \beta + 66) q^{85} + ( - 36 \beta + 378) q^{87} + ( - 71 \beta + 183) q^{89} + ( - 42 \beta + 56) q^{91} + ( - 108 \beta - 60) q^{93} + (4 \beta + 588) q^{95} + (138 \beta + 404) q^{97} + (45 \beta - 27) q^{99}+O(q^{100})$$ q + 3 * q^3 + (-b - 3) * q^5 - 7 * q^7 + 9 * q^9 + (5*b - 3) * q^11 + (6*b - 8) * q^13 + (-3*b - 9) * q^15 + (-b - 3) * q^17 + (-12*b + 32) * q^19 - 21 * q^21 + (17*b - 3) * q^23 + (6*b - 59) * q^25 + 27 * q^27 + (-12*b + 126) * q^29 + (-36*b - 20) * q^31 + (15*b - 9) * q^33 + (7*b + 21) * q^35 + (18*b + 124) * q^37 + (18*b - 24) * q^39 + (-15*b - 225) * q^41 + (-24*b + 188) * q^43 + (-9*b - 27) * q^45 + (-34*b + 6) * q^47 + 49 * q^49 + (-3*b - 9) * q^51 + (-2*b + 552) * q^53 + (-12*b - 276) * q^55 + (-36*b + 96) * q^57 + (-58*b + 402) * q^59 + (-36*b + 214) * q^61 - 63 * q^63 + (-10*b - 318) * q^65 + (54*b + 74) * q^67 + (51*b - 9) * q^69 + (15*b - 477) * q^71 + (6*b + 536) * q^73 + (18*b - 177) * q^75 + (-35*b + 21) * q^77 + (54*b + 286) * q^79 + 81 * q^81 + (48*b + 972) * q^83 + (6*b + 66) * q^85 + (-36*b + 378) * q^87 + (-71*b + 183) * q^89 + (-42*b + 56) * q^91 + (-108*b - 60) * q^93 + (4*b + 588) * q^95 + (138*b + 404) * q^97 + (45*b - 27) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 6 q^{3} - 6 q^{5} - 14 q^{7} + 18 q^{9}+O(q^{10})$$ 2 * q + 6 * q^3 - 6 * q^5 - 14 * q^7 + 18 * q^9 $$2 q + 6 q^{3} - 6 q^{5} - 14 q^{7} + 18 q^{9} - 6 q^{11} - 16 q^{13} - 18 q^{15} - 6 q^{17} + 64 q^{19} - 42 q^{21} - 6 q^{23} - 118 q^{25} + 54 q^{27} + 252 q^{29} - 40 q^{31} - 18 q^{33} + 42 q^{35} + 248 q^{37} - 48 q^{39} - 450 q^{41} + 376 q^{43} - 54 q^{45} + 12 q^{47} + 98 q^{49} - 18 q^{51} + 1104 q^{53} - 552 q^{55} + 192 q^{57} + 804 q^{59} + 428 q^{61} - 126 q^{63} - 636 q^{65} + 148 q^{67} - 18 q^{69} - 954 q^{71} + 1072 q^{73} - 354 q^{75} + 42 q^{77} + 572 q^{79} + 162 q^{81} + 1944 q^{83} + 132 q^{85} + 756 q^{87} + 366 q^{89} + 112 q^{91} - 120 q^{93} + 1176 q^{95} + 808 q^{97} - 54 q^{99}+O(q^{100})$$ 2 * q + 6 * q^3 - 6 * q^5 - 14 * q^7 + 18 * q^9 - 6 * q^11 - 16 * q^13 - 18 * q^15 - 6 * q^17 + 64 * q^19 - 42 * q^21 - 6 * q^23 - 118 * q^25 + 54 * q^27 + 252 * q^29 - 40 * q^31 - 18 * q^33 + 42 * q^35 + 248 * q^37 - 48 * q^39 - 450 * q^41 + 376 * q^43 - 54 * q^45 + 12 * q^47 + 98 * q^49 - 18 * q^51 + 1104 * q^53 - 552 * q^55 + 192 * q^57 + 804 * q^59 + 428 * q^61 - 126 * q^63 - 636 * q^65 + 148 * q^67 - 18 * q^69 - 954 * q^71 + 1072 * q^73 - 354 * q^75 + 42 * q^77 + 572 * q^79 + 162 * q^81 + 1944 * q^83 + 132 * q^85 + 756 * q^87 + 366 * q^89 + 112 * q^91 - 120 * q^93 + 1176 * q^95 + 808 * q^97 - 54 * 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
 4.27492 −3.27492
0 3.00000 0 −10.5498 0 −7.00000 0 9.00000 0
1.2 0 3.00000 0 4.54983 0 −7.00000 0 9.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$7$$ $$1$$

## 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 1344.4.a.bo 2
4.b odd 2 1 1344.4.a.bg 2
8.b even 2 1 336.4.a.m 2
8.d odd 2 1 21.4.a.c 2
24.f even 2 1 63.4.a.e 2
24.h odd 2 1 1008.4.a.ba 2
40.e odd 2 1 525.4.a.n 2
40.k even 4 2 525.4.d.g 4
56.e even 2 1 147.4.a.i 2
56.h odd 2 1 2352.4.a.bz 2
56.k odd 6 2 147.4.e.l 4
56.m even 6 2 147.4.e.m 4
120.m even 2 1 1575.4.a.p 2
168.e odd 2 1 441.4.a.r 2
168.v even 6 2 441.4.e.q 4
168.be odd 6 2 441.4.e.p 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.a.c 2 8.d odd 2 1
63.4.a.e 2 24.f even 2 1
147.4.a.i 2 56.e even 2 1
147.4.e.l 4 56.k odd 6 2
147.4.e.m 4 56.m even 6 2
336.4.a.m 2 8.b even 2 1
441.4.a.r 2 168.e odd 2 1
441.4.e.p 4 168.be odd 6 2
441.4.e.q 4 168.v even 6 2
525.4.a.n 2 40.e odd 2 1
525.4.d.g 4 40.k even 4 2
1008.4.a.ba 2 24.h odd 2 1
1344.4.a.bg 2 4.b odd 2 1
1344.4.a.bo 2 1.a even 1 1 trivial
1575.4.a.p 2 120.m even 2 1
2352.4.a.bz 2 56.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1344))$$:

 $$T_{5}^{2} + 6T_{5} - 48$$ T5^2 + 6*T5 - 48 $$T_{11}^{2} + 6T_{11} - 1416$$ T11^2 + 6*T11 - 1416

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T - 3)^{2}$$
$5$ $$T^{2} + 6T - 48$$
$7$ $$(T + 7)^{2}$$
$11$ $$T^{2} + 6T - 1416$$
$13$ $$T^{2} + 16T - 1988$$
$17$ $$T^{2} + 6T - 48$$
$19$ $$T^{2} - 64T - 7184$$
$23$ $$T^{2} + 6T - 16464$$
$29$ $$T^{2} - 252T + 7668$$
$31$ $$T^{2} + 40T - 73472$$
$37$ $$T^{2} - 248T - 3092$$
$41$ $$T^{2} + 450T + 37800$$
$43$ $$T^{2} - 376T + 2512$$
$47$ $$T^{2} - 12T - 65856$$
$53$ $$T^{2} - 1104 T + 304476$$
$59$ $$T^{2} - 804T - 30144$$
$61$ $$T^{2} - 428T - 28076$$
$67$ $$T^{2} - 148T - 160736$$
$71$ $$T^{2} + 954T + 214704$$
$73$ $$T^{2} - 1072 T + 285244$$
$79$ $$T^{2} - 572T - 84416$$
$83$ $$T^{2} - 1944 T + 813456$$
$89$ $$T^{2} - 366T - 253848$$
$97$ $$T^{2} - 808T - 922292$$