# Properties

 Label 2112.4.a.bl Level $2112$ Weight $4$ Character orbit 2112.a Self dual yes Analytic conductor $124.612$ 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] = [2112,4,Mod(1,2112)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2112, 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("2112.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2112 = 2^{6} \cdot 3 \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2112.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: $$124.612033932$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 264) 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 $$\beta = \sqrt{17}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 3 q^{3} + (\beta + 3) q^{5} + (\beta + 5) q^{7} + 9 q^{9}+O(q^{10})$$ q + 3 * q^3 + (b + 3) * q^5 + (b + 5) * q^7 + 9 * q^9 $$q + 3 q^{3} + (\beta + 3) q^{5} + (\beta + 5) q^{7} + 9 q^{9} - 11 q^{11} + ( - 17 \beta + 1) q^{13} + (3 \beta + 9) q^{15} + ( - 18 \beta - 52) q^{17} + (30 \beta - 2) q^{19} + (3 \beta + 15) q^{21} + (13 \beta - 51) q^{23} + (6 \beta - 99) q^{25} + 27 q^{27} + ( - 26 \beta + 196) q^{29} + (40 \beta - 32) q^{31} - 33 q^{33} + (8 \beta + 32) q^{35} + (80 \beta + 82) q^{37} + ( - 51 \beta + 3) q^{39} + (20 \beta - 366) q^{41} + ( - 68 \beta - 84) q^{43} + (9 \beta + 27) q^{45} + ( - 45 \beta - 157) q^{47} + (10 \beta - 301) q^{49} + ( - 54 \beta - 156) q^{51} + (37 \beta + 191) q^{53} + ( - 11 \beta - 33) q^{55} + (90 \beta - 6) q^{57} + ( - 58 \beta - 254) q^{59} + ( - 151 \beta + 3) q^{61} + (9 \beta + 45) q^{63} + ( - 50 \beta - 286) q^{65} + (136 \beta - 108) q^{67} + (39 \beta - 153) q^{69} + ( - 63 \beta - 439) q^{71} + ( - 224 \beta + 130) q^{73} + (18 \beta - 297) q^{75} + ( - 11 \beta - 55) q^{77} + ( - 97 \beta + 59) q^{79} + 81 q^{81} + ( - 212 \beta - 248) q^{83} + ( - 106 \beta - 462) q^{85} + ( - 78 \beta + 588) q^{87} + (168 \beta - 878) q^{89} + ( - 84 \beta - 284) q^{91} + (120 \beta - 96) q^{93} + (88 \beta + 504) q^{95} + (38 \beta + 984) q^{97} - 99 q^{99}+O(q^{100})$$ q + 3 * q^3 + (b + 3) * q^5 + (b + 5) * q^7 + 9 * q^9 - 11 * q^11 + (-17*b + 1) * q^13 + (3*b + 9) * q^15 + (-18*b - 52) * q^17 + (30*b - 2) * q^19 + (3*b + 15) * q^21 + (13*b - 51) * q^23 + (6*b - 99) * q^25 + 27 * q^27 + (-26*b + 196) * q^29 + (40*b - 32) * q^31 - 33 * q^33 + (8*b + 32) * q^35 + (80*b + 82) * q^37 + (-51*b + 3) * q^39 + (20*b - 366) * q^41 + (-68*b - 84) * q^43 + (9*b + 27) * q^45 + (-45*b - 157) * q^47 + (10*b - 301) * q^49 + (-54*b - 156) * q^51 + (37*b + 191) * q^53 + (-11*b - 33) * q^55 + (90*b - 6) * q^57 + (-58*b - 254) * q^59 + (-151*b + 3) * q^61 + (9*b + 45) * q^63 + (-50*b - 286) * q^65 + (136*b - 108) * q^67 + (39*b - 153) * q^69 + (-63*b - 439) * q^71 + (-224*b + 130) * q^73 + (18*b - 297) * q^75 + (-11*b - 55) * q^77 + (-97*b + 59) * q^79 + 81 * q^81 + (-212*b - 248) * q^83 + (-106*b - 462) * q^85 + (-78*b + 588) * q^87 + (168*b - 878) * q^89 + (-84*b - 284) * q^91 + (120*b - 96) * q^93 + (88*b + 504) * q^95 + (38*b + 984) * q^97 - 99 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 6 q^{3} + 6 q^{5} + 10 q^{7} + 18 q^{9}+O(q^{10})$$ 2 * q + 6 * q^3 + 6 * q^5 + 10 * q^7 + 18 * q^9 $$2 q + 6 q^{3} + 6 q^{5} + 10 q^{7} + 18 q^{9} - 22 q^{11} + 2 q^{13} + 18 q^{15} - 104 q^{17} - 4 q^{19} + 30 q^{21} - 102 q^{23} - 198 q^{25} + 54 q^{27} + 392 q^{29} - 64 q^{31} - 66 q^{33} + 64 q^{35} + 164 q^{37} + 6 q^{39} - 732 q^{41} - 168 q^{43} + 54 q^{45} - 314 q^{47} - 602 q^{49} - 312 q^{51} + 382 q^{53} - 66 q^{55} - 12 q^{57} - 508 q^{59} + 6 q^{61} + 90 q^{63} - 572 q^{65} - 216 q^{67} - 306 q^{69} - 878 q^{71} + 260 q^{73} - 594 q^{75} - 110 q^{77} + 118 q^{79} + 162 q^{81} - 496 q^{83} - 924 q^{85} + 1176 q^{87} - 1756 q^{89} - 568 q^{91} - 192 q^{93} + 1008 q^{95} + 1968 q^{97} - 198 q^{99}+O(q^{100})$$ 2 * q + 6 * q^3 + 6 * q^5 + 10 * q^7 + 18 * q^9 - 22 * q^11 + 2 * q^13 + 18 * q^15 - 104 * q^17 - 4 * q^19 + 30 * q^21 - 102 * q^23 - 198 * q^25 + 54 * q^27 + 392 * q^29 - 64 * q^31 - 66 * q^33 + 64 * q^35 + 164 * q^37 + 6 * q^39 - 732 * q^41 - 168 * q^43 + 54 * q^45 - 314 * q^47 - 602 * q^49 - 312 * q^51 + 382 * q^53 - 66 * q^55 - 12 * q^57 - 508 * q^59 + 6 * q^61 + 90 * q^63 - 572 * q^65 - 216 * q^67 - 306 * q^69 - 878 * q^71 + 260 * q^73 - 594 * q^75 - 110 * q^77 + 118 * q^79 + 162 * q^81 - 496 * q^83 - 924 * q^85 + 1176 * q^87 - 1756 * q^89 - 568 * q^91 - 192 * q^93 + 1008 * q^95 + 1968 * q^97 - 198 * 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
 −1.56155 2.56155
0 3.00000 0 −1.12311 0 0.876894 0 9.00000 0
1.2 0 3.00000 0 7.12311 0 9.12311 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
$$2$$ $$+1$$
$$3$$ $$-1$$
$$11$$ $$+1$$

## 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 2112.4.a.bl 2
4.b odd 2 1 2112.4.a.be 2
8.b even 2 1 264.4.a.e 2
8.d odd 2 1 528.4.a.r 2
24.f even 2 1 1584.4.a.bf 2
24.h odd 2 1 792.4.a.h 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
264.4.a.e 2 8.b even 2 1
528.4.a.r 2 8.d odd 2 1
792.4.a.h 2 24.h odd 2 1
1584.4.a.bf 2 24.f even 2 1
2112.4.a.be 2 4.b odd 2 1
2112.4.a.bl 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(2112))$$:

 $$T_{5}^{2} - 6T_{5} - 8$$ T5^2 - 6*T5 - 8 $$T_{7}^{2} - 10T_{7} + 8$$ T7^2 - 10*T7 + 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T - 3)^{2}$$
$5$ $$T^{2} - 6T - 8$$
$7$ $$T^{2} - 10T + 8$$
$11$ $$(T + 11)^{2}$$
$13$ $$T^{2} - 2T - 4912$$
$17$ $$T^{2} + 104T - 2804$$
$19$ $$T^{2} + 4T - 15296$$
$23$ $$T^{2} + 102T - 272$$
$29$ $$T^{2} - 392T + 26924$$
$31$ $$T^{2} + 64T - 26176$$
$37$ $$T^{2} - 164T - 102076$$
$41$ $$T^{2} + 732T + 127156$$
$43$ $$T^{2} + 168T - 71552$$
$47$ $$T^{2} + 314T - 9776$$
$53$ $$T^{2} - 382T + 13208$$
$59$ $$T^{2} + 508T + 7328$$
$61$ $$T^{2} - 6T - 387608$$
$67$ $$T^{2} + 216T - 302768$$
$71$ $$T^{2} + 878T + 125248$$
$73$ $$T^{2} - 260T - 836092$$
$79$ $$T^{2} - 118T - 156472$$
$83$ $$T^{2} + 496T - 702544$$
$89$ $$T^{2} + 1756 T + 291076$$
$97$ $$T^{2} - 1968 T + 943708$$