# Properties

 Label 2352.4.a.bz Level $2352$ Weight $4$ Character orbit 2352.a Self dual yes Analytic conductor $138.772$ Analytic rank $1$ 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] = [2352,4,Mod(1,2352)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2352, 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("2352.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2352 = 2^{4} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2352.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: $$138.772492334$$ Analytic rank: $$1$$ 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.

 $$f(q)$$ $$=$$ $$q + 3 q^{3} + ( - \beta - 3) q^{5} + 9 q^{9}+O(q^{10})$$ q + 3 * q^3 + (-b - 3) * q^5 + 9 * q^9 $$q + 3 q^{3} + ( - \beta - 3) q^{5} + 9 q^{9} + ( - 5 \beta + 3) q^{11} + (6 \beta - 8) q^{13} + ( - 3 \beta - 9) q^{15} + (\beta + 3) q^{17} + ( - 12 \beta + 32) q^{19} + (17 \beta - 3) q^{23} + (6 \beta - 59) q^{25} + 27 q^{27} + (12 \beta - 126) q^{29} + (36 \beta + 20) q^{31} + ( - 15 \beta + 9) q^{33} + ( - 18 \beta - 124) q^{37} + (18 \beta - 24) q^{39} + (15 \beta + 225) q^{41} + (24 \beta - 188) q^{43} + ( - 9 \beta - 27) q^{45} + (34 \beta - 6) q^{47} + (3 \beta + 9) q^{51} + (2 \beta - 552) q^{53} + (12 \beta + 276) q^{55} + ( - 36 \beta + 96) q^{57} + ( - 58 \beta + 402) q^{59} + ( - 36 \beta + 214) q^{61} + ( - 10 \beta - 318) q^{65} + ( - 54 \beta - 74) q^{67} + (51 \beta - 9) q^{69} + (15 \beta - 477) q^{71} + ( - 6 \beta - 536) q^{73} + (18 \beta - 177) q^{75} + (54 \beta + 286) q^{79} + 81 q^{81} + (48 \beta + 972) q^{83} + ( - 6 \beta - 66) q^{85} + (36 \beta - 378) q^{87} + (71 \beta - 183) q^{89} + (108 \beta + 60) q^{93} + (4 \beta + 588) q^{95} + ( - 138 \beta - 404) q^{97} + ( - 45 \beta + 27) q^{99}+O(q^{100})$$ q + 3 * q^3 + (-b - 3) * q^5 + 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 + (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 + (-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 + (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 + (-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 + (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 + (108*b + 60) * q^93 + (4*b + 588) * q^95 + (-138*b - 404) * q^97 + (-45*b + 27) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 6 q^{3} - 6 q^{5} + 18 q^{9}+O(q^{10})$$ 2 * q + 6 * q^3 - 6 * q^5 + 18 * q^9 $$2 q + 6 q^{3} - 6 q^{5} + 18 q^{9} + 6 q^{11} - 16 q^{13} - 18 q^{15} + 6 q^{17} + 64 q^{19} - 6 q^{23} - 118 q^{25} + 54 q^{27} - 252 q^{29} + 40 q^{31} + 18 q^{33} - 248 q^{37} - 48 q^{39} + 450 q^{41} - 376 q^{43} - 54 q^{45} - 12 q^{47} + 18 q^{51} - 1104 q^{53} + 552 q^{55} + 192 q^{57} + 804 q^{59} + 428 q^{61} - 636 q^{65} - 148 q^{67} - 18 q^{69} - 954 q^{71} - 1072 q^{73} - 354 q^{75} + 572 q^{79} + 162 q^{81} + 1944 q^{83} - 132 q^{85} - 756 q^{87} - 366 q^{89} + 120 q^{93} + 1176 q^{95} - 808 q^{97} + 54 q^{99}+O(q^{100})$$ 2 * q + 6 * q^3 - 6 * q^5 + 18 * q^9 + 6 * q^11 - 16 * q^13 - 18 * q^15 + 6 * q^17 + 64 * q^19 - 6 * q^23 - 118 * q^25 + 54 * q^27 - 252 * q^29 + 40 * q^31 + 18 * q^33 - 248 * q^37 - 48 * q^39 + 450 * q^41 - 376 * q^43 - 54 * q^45 - 12 * q^47 + 18 * q^51 - 1104 * q^53 + 552 * q^55 + 192 * q^57 + 804 * q^59 + 428 * q^61 - 636 * q^65 - 148 * q^67 - 18 * q^69 - 954 * q^71 - 1072 * q^73 - 354 * q^75 + 572 * q^79 + 162 * q^81 + 1944 * q^83 - 132 * q^85 - 756 * q^87 - 366 * q^89 + 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 0 0 9.00000 0
1.2 0 3.00000 0 4.54983 0 0 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 2352.4.a.bz 2
4.b odd 2 1 147.4.a.i 2
7.b odd 2 1 336.4.a.m 2
12.b even 2 1 441.4.a.r 2
21.c even 2 1 1008.4.a.ba 2
28.d even 2 1 21.4.a.c 2
28.f even 6 2 147.4.e.l 4
28.g odd 6 2 147.4.e.m 4
56.e even 2 1 1344.4.a.bg 2
56.h odd 2 1 1344.4.a.bo 2
84.h odd 2 1 63.4.a.e 2
84.j odd 6 2 441.4.e.q 4
84.n even 6 2 441.4.e.p 4
140.c even 2 1 525.4.a.n 2
140.j odd 4 2 525.4.d.g 4
420.o odd 2 1 1575.4.a.p 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.a.c 2 28.d even 2 1
63.4.a.e 2 84.h odd 2 1
147.4.a.i 2 4.b odd 2 1
147.4.e.l 4 28.f even 6 2
147.4.e.m 4 28.g odd 6 2
336.4.a.m 2 7.b odd 2 1
441.4.a.r 2 12.b even 2 1
441.4.e.p 4 84.n even 6 2
441.4.e.q 4 84.j odd 6 2
525.4.a.n 2 140.c even 2 1
525.4.d.g 4 140.j odd 4 2
1008.4.a.ba 2 21.c even 2 1
1344.4.a.bg 2 56.e even 2 1
1344.4.a.bo 2 56.h odd 2 1
1575.4.a.p 2 420.o odd 2 1
2352.4.a.bz 2 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(2352))$$:

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

## Hecke characteristic polynomials

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