# Properties

 Label 112.6.a.h Level $112$ Weight $6$ Character orbit 112.a Self dual yes Analytic conductor $17.963$ 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] = [112,6,Mod(1,112)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(112, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("112.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$112 = 2^{4} \cdot 7$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 112.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: $$17.9629878191$$ 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 7) 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.

 $$f(q)$$ $$=$$ $$q + (3 \beta + 3) q^{3} + (5 \beta - 9) q^{5} - 49 q^{7} + (18 \beta + 279) q^{9}+O(q^{10})$$ q + (3*b + 3) * q^3 + (5*b - 9) * q^5 - 49 * q^7 + (18*b + 279) * q^9 $$q + (3 \beta + 3) q^{3} + (5 \beta - 9) q^{5} - 49 q^{7} + (18 \beta + 279) q^{9} + (62 \beta - 198) q^{11} + (63 \beta - 175) q^{13} + ( - 12 \beta + 828) q^{15} + ( - 38 \beta + 900) q^{17} + (9 \beta + 1633) q^{19} + ( - 147 \beta - 147) q^{21} + ( - 284 \beta - 1044) q^{23} + ( - 90 \beta - 1619) q^{25} + (162 \beta + 3186) q^{27} + (126 \beta + 3348) q^{29} + ( - 270 \beta + 10) q^{31} + ( - 408 \beta + 10008) q^{33} + ( - 245 \beta + 441) q^{35} + (270 \beta + 3116) q^{37} + ( - 336 \beta + 10248) q^{39} + ( - 546 \beta - 3024) q^{41} + (2394 \beta + 1510) q^{43} + (1233 \beta + 2619) q^{45} + ( - 1874 \beta - 5850) q^{47} + 2401 q^{49} + (2586 \beta - 3798) q^{51} + ( - 104 \beta + 4734) q^{53} + ( - 1548 \beta + 19452) q^{55} + (4926 \beta + 6438) q^{57} + (1025 \beta + 21969) q^{59} + ( - 2403 \beta - 32377) q^{61} + ( - 882 \beta - 13671) q^{63} + ( - 1442 \beta + 19530) q^{65} + (972 \beta - 12392) q^{67} + ( - 3984 \beta - 51696) q^{69} + (2100 \beta - 48708) q^{71} + ( - 2628 \beta + 8726) q^{73} + ( - 5127 \beta - 20247) q^{75} + ( - 3038 \beta + 9702) q^{77} + ( - 7452 \beta - 25628) q^{79} + (5670 \beta - 30537) q^{81} + ( - 7875 \beta - 58779) q^{83} + (4842 \beta - 18930) q^{85} + (10422 \beta + 31590) q^{87} + ( - 11104 \beta + 42138) q^{89} + ( - 3087 \beta + 8575) q^{91} + ( - 780 \beta - 46140) q^{93} + (8084 \beta - 12132) q^{95} + (4410 \beta + 10388) q^{97} + (13734 \beta + 8370) q^{99}+O(q^{100})$$ q + (3*b + 3) * q^3 + (5*b - 9) * q^5 - 49 * q^7 + (18*b + 279) * q^9 + (62*b - 198) * q^11 + (63*b - 175) * q^13 + (-12*b + 828) * q^15 + (-38*b + 900) * q^17 + (9*b + 1633) * q^19 + (-147*b - 147) * q^21 + (-284*b - 1044) * q^23 + (-90*b - 1619) * q^25 + (162*b + 3186) * q^27 + (126*b + 3348) * q^29 + (-270*b + 10) * q^31 + (-408*b + 10008) * q^33 + (-245*b + 441) * q^35 + (270*b + 3116) * q^37 + (-336*b + 10248) * q^39 + (-546*b - 3024) * q^41 + (2394*b + 1510) * q^43 + (1233*b + 2619) * q^45 + (-1874*b - 5850) * q^47 + 2401 * q^49 + (2586*b - 3798) * q^51 + (-104*b + 4734) * q^53 + (-1548*b + 19452) * q^55 + (4926*b + 6438) * q^57 + (1025*b + 21969) * q^59 + (-2403*b - 32377) * q^61 + (-882*b - 13671) * q^63 + (-1442*b + 19530) * q^65 + (972*b - 12392) * q^67 + (-3984*b - 51696) * q^69 + (2100*b - 48708) * q^71 + (-2628*b + 8726) * q^73 + (-5127*b - 20247) * q^75 + (-3038*b + 9702) * q^77 + (-7452*b - 25628) * q^79 + (5670*b - 30537) * q^81 + (-7875*b - 58779) * q^83 + (4842*b - 18930) * q^85 + (10422*b + 31590) * q^87 + (-11104*b + 42138) * q^89 + (-3087*b + 8575) * q^91 + (-780*b - 46140) * q^93 + (8084*b - 12132) * q^95 + (4410*b + 10388) * q^97 + (13734*b + 8370) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 6 q^{3} - 18 q^{5} - 98 q^{7} + 558 q^{9}+O(q^{10})$$ 2 * q + 6 * q^3 - 18 * q^5 - 98 * q^7 + 558 * q^9 $$2 q + 6 q^{3} - 18 q^{5} - 98 q^{7} + 558 q^{9} - 396 q^{11} - 350 q^{13} + 1656 q^{15} + 1800 q^{17} + 3266 q^{19} - 294 q^{21} - 2088 q^{23} - 3238 q^{25} + 6372 q^{27} + 6696 q^{29} + 20 q^{31} + 20016 q^{33} + 882 q^{35} + 6232 q^{37} + 20496 q^{39} - 6048 q^{41} + 3020 q^{43} + 5238 q^{45} - 11700 q^{47} + 4802 q^{49} - 7596 q^{51} + 9468 q^{53} + 38904 q^{55} + 12876 q^{57} + 43938 q^{59} - 64754 q^{61} - 27342 q^{63} + 39060 q^{65} - 24784 q^{67} - 103392 q^{69} - 97416 q^{71} + 17452 q^{73} - 40494 q^{75} + 19404 q^{77} - 51256 q^{79} - 61074 q^{81} - 117558 q^{83} - 37860 q^{85} + 63180 q^{87} + 84276 q^{89} + 17150 q^{91} - 92280 q^{93} - 24264 q^{95} + 20776 q^{97} + 16740 q^{99}+O(q^{100})$$ 2 * q + 6 * q^3 - 18 * q^5 - 98 * q^7 + 558 * q^9 - 396 * q^11 - 350 * q^13 + 1656 * q^15 + 1800 * q^17 + 3266 * q^19 - 294 * q^21 - 2088 * q^23 - 3238 * q^25 + 6372 * q^27 + 6696 * q^29 + 20 * q^31 + 20016 * q^33 + 882 * q^35 + 6232 * q^37 + 20496 * q^39 - 6048 * q^41 + 3020 * q^43 + 5238 * q^45 - 11700 * q^47 + 4802 * q^49 - 7596 * q^51 + 9468 * q^53 + 38904 * q^55 + 12876 * q^57 + 43938 * q^59 - 64754 * q^61 - 27342 * q^63 + 39060 * q^65 - 24784 * q^67 - 103392 * q^69 - 97416 * q^71 + 17452 * q^73 - 40494 * q^75 + 19404 * q^77 - 51256 * q^79 - 61074 * q^81 - 117558 * q^83 - 37860 * q^85 + 63180 * q^87 + 84276 * q^89 + 17150 * q^91 - 92280 * q^93 - 24264 * q^95 + 20776 * q^97 + 16740 * 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
 −3.27492 4.27492
0 −19.6495 0 −46.7492 0 −49.0000 0 143.103 0
1.2 0 25.6495 0 28.7492 0 −49.0000 0 414.897 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
$$2$$ $$-1$$
$$7$$ $$+1$$

## 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 112.6.a.h 2
3.b odd 2 1 1008.6.a.bq 2
4.b odd 2 1 7.6.a.b 2
7.b odd 2 1 784.6.a.v 2
8.b even 2 1 448.6.a.u 2
8.d odd 2 1 448.6.a.w 2
12.b even 2 1 63.6.a.f 2
20.d odd 2 1 175.6.a.c 2
20.e even 4 2 175.6.b.c 4
28.d even 2 1 49.6.a.f 2
28.f even 6 2 49.6.c.d 4
28.g odd 6 2 49.6.c.e 4
44.c even 2 1 847.6.a.c 2
84.h odd 2 1 441.6.a.l 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.6.a.b 2 4.b odd 2 1
49.6.a.f 2 28.d even 2 1
49.6.c.d 4 28.f even 6 2
49.6.c.e 4 28.g odd 6 2
63.6.a.f 2 12.b even 2 1
112.6.a.h 2 1.a even 1 1 trivial
175.6.a.c 2 20.d odd 2 1
175.6.b.c 4 20.e even 4 2
441.6.a.l 2 84.h odd 2 1
448.6.a.u 2 8.b even 2 1
448.6.a.w 2 8.d odd 2 1
784.6.a.v 2 7.b odd 2 1
847.6.a.c 2 44.c even 2 1
1008.6.a.bq 2 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - 6T_{3} - 504$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(112))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 6T - 504$$
$5$ $$T^{2} + 18T - 1344$$
$7$ $$(T + 49)^{2}$$
$11$ $$T^{2} + 396T - 179904$$
$13$ $$T^{2} + 350T - 195608$$
$17$ $$T^{2} - 1800 T + 727692$$
$19$ $$T^{2} - 3266 T + 2662072$$
$23$ $$T^{2} + 2088 T - 3507456$$
$29$ $$T^{2} - 6696 T + 10304172$$
$31$ $$T^{2} - 20T - 4155200$$
$37$ $$T^{2} - 6232 T + 5554156$$
$41$ $$T^{2} + 6048 T - 7848036$$
$43$ $$T^{2} - 3020 T - 324400352$$
$47$ $$T^{2} + 11700 T - 165954432$$
$53$ $$T^{2} - 9468 T + 21794244$$
$59$ $$T^{2} - 43938 T + 422751336$$
$61$ $$T^{2} + 64754 T + 719128816$$
$67$ $$T^{2} + 24784 T + 99708976$$
$71$ $$T^{2} + \cdots + 2121099264$$
$73$ $$T^{2} - 17452 T - 317520812$$
$79$ $$T^{2} + \cdots - 2508546944$$
$83$ $$T^{2} + 117558 T - 79919784$$
$89$ $$T^{2} + \cdots - 5252421468$$
$97$ $$T^{2} + \cdots - 1000631156$$