# Properties

 Label 117.4.a.c Level $117$ Weight $4$ Character orbit 117.a Self dual yes Analytic conductor $6.903$ 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] = [117,4,Mod(1,117)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(117, 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("117.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$117 = 3^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 117.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: $$6.90322347067$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{14})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 14$$ x^2 - 14 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 39) 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 $$\beta = \sqrt{14}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta - 1) q^{2} + ( - 2 \beta + 7) q^{4} + ( - 2 \beta - 12) q^{5} + 2 \beta q^{7} + (\beta - 27) q^{8}+O(q^{10})$$ q + (b - 1) * q^2 + (-2*b + 7) * q^4 + (-2*b - 12) * q^5 + 2*b * q^7 + (b - 27) * q^8 $$q + (\beta - 1) q^{2} + ( - 2 \beta + 7) q^{4} + ( - 2 \beta - 12) q^{5} + 2 \beta q^{7} + (\beta - 27) q^{8} + ( - 10 \beta - 16) q^{10} + ( - 12 \beta + 22) q^{11} - 13 q^{13} + ( - 2 \beta + 28) q^{14} + ( - 12 \beta - 15) q^{16} + (4 \beta - 82) q^{17} + ( - 2 \beta + 24) q^{19} + (10 \beta - 28) q^{20} + (34 \beta - 190) q^{22} + (48 \beta - 4) q^{23} + (48 \beta + 75) q^{25} + ( - 13 \beta + 13) q^{26} + (14 \beta - 56) q^{28} + ( - 24 \beta - 202) q^{29} + (26 \beta + 20) q^{31} + ( - 11 \beta + 63) q^{32} + ( - 86 \beta + 138) q^{34} + ( - 24 \beta - 56) q^{35} + ( - 28 \beta - 50) q^{37} + (26 \beta - 52) q^{38} + (42 \beta + 296) q^{40} + (94 \beta - 100) q^{41} + ( - 52 \beta - 308) q^{43} + ( - 128 \beta + 490) q^{44} + ( - 52 \beta + 676) q^{46} + (32 \beta + 162) q^{47} - 287 q^{49} + (27 \beta + 597) q^{50} + (26 \beta - 91) q^{52} + ( - 120 \beta + 82) q^{53} + (100 \beta + 72) q^{55} + ( - 54 \beta + 28) q^{56} + ( - 178 \beta - 134) q^{58} + (40 \beta - 70) q^{59} + ( - 136 \beta + 314) q^{61} + ( - 6 \beta + 344) q^{62} + (170 \beta - 97) q^{64} + (26 \beta + 156) q^{65} + (170 \beta - 236) q^{67} + (192 \beta - 686) q^{68} + ( - 32 \beta - 280) q^{70} + ( - 84 \beta - 214) q^{71} + ( - 76 \beta - 450) q^{73} + ( - 22 \beta - 342) q^{74} + ( - 62 \beta + 224) q^{76} + (44 \beta - 336) q^{77} + (88 \beta - 216) q^{79} + (174 \beta + 516) q^{80} + ( - 194 \beta + 1416) q^{82} + (64 \beta + 694) q^{83} + (116 \beta + 872) q^{85} + ( - 256 \beta - 420) q^{86} + (346 \beta - 762) q^{88} + ( - 190 \beta - 480) q^{89} - 26 \beta q^{91} + (344 \beta - 1372) q^{92} + (130 \beta + 286) q^{94} + ( - 24 \beta - 232) q^{95} + (220 \beta - 266) q^{97} + ( - 287 \beta + 287) q^{98} +O(q^{100})$$ q + (b - 1) * q^2 + (-2*b + 7) * q^4 + (-2*b - 12) * q^5 + 2*b * q^7 + (b - 27) * q^8 + (-10*b - 16) * q^10 + (-12*b + 22) * q^11 - 13 * q^13 + (-2*b + 28) * q^14 + (-12*b - 15) * q^16 + (4*b - 82) * q^17 + (-2*b + 24) * q^19 + (10*b - 28) * q^20 + (34*b - 190) * q^22 + (48*b - 4) * q^23 + (48*b + 75) * q^25 + (-13*b + 13) * q^26 + (14*b - 56) * q^28 + (-24*b - 202) * q^29 + (26*b + 20) * q^31 + (-11*b + 63) * q^32 + (-86*b + 138) * q^34 + (-24*b - 56) * q^35 + (-28*b - 50) * q^37 + (26*b - 52) * q^38 + (42*b + 296) * q^40 + (94*b - 100) * q^41 + (-52*b - 308) * q^43 + (-128*b + 490) * q^44 + (-52*b + 676) * q^46 + (32*b + 162) * q^47 - 287 * q^49 + (27*b + 597) * q^50 + (26*b - 91) * q^52 + (-120*b + 82) * q^53 + (100*b + 72) * q^55 + (-54*b + 28) * q^56 + (-178*b - 134) * q^58 + (40*b - 70) * q^59 + (-136*b + 314) * q^61 + (-6*b + 344) * q^62 + (170*b - 97) * q^64 + (26*b + 156) * q^65 + (170*b - 236) * q^67 + (192*b - 686) * q^68 + (-32*b - 280) * q^70 + (-84*b - 214) * q^71 + (-76*b - 450) * q^73 + (-22*b - 342) * q^74 + (-62*b + 224) * q^76 + (44*b - 336) * q^77 + (88*b - 216) * q^79 + (174*b + 516) * q^80 + (-194*b + 1416) * q^82 + (64*b + 694) * q^83 + (116*b + 872) * q^85 + (-256*b - 420) * q^86 + (346*b - 762) * q^88 + (-190*b - 480) * q^89 - 26*b * q^91 + (344*b - 1372) * q^92 + (130*b + 286) * q^94 + (-24*b - 232) * q^95 + (220*b - 266) * q^97 + (-287*b + 287) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 14 q^{4} - 24 q^{5} - 54 q^{8}+O(q^{10})$$ 2 * q - 2 * q^2 + 14 * q^4 - 24 * q^5 - 54 * q^8 $$2 q - 2 q^{2} + 14 q^{4} - 24 q^{5} - 54 q^{8} - 32 q^{10} + 44 q^{11} - 26 q^{13} + 56 q^{14} - 30 q^{16} - 164 q^{17} + 48 q^{19} - 56 q^{20} - 380 q^{22} - 8 q^{23} + 150 q^{25} + 26 q^{26} - 112 q^{28} - 404 q^{29} + 40 q^{31} + 126 q^{32} + 276 q^{34} - 112 q^{35} - 100 q^{37} - 104 q^{38} + 592 q^{40} - 200 q^{41} - 616 q^{43} + 980 q^{44} + 1352 q^{46} + 324 q^{47} - 574 q^{49} + 1194 q^{50} - 182 q^{52} + 164 q^{53} + 144 q^{55} + 56 q^{56} - 268 q^{58} - 140 q^{59} + 628 q^{61} + 688 q^{62} - 194 q^{64} + 312 q^{65} - 472 q^{67} - 1372 q^{68} - 560 q^{70} - 428 q^{71} - 900 q^{73} - 684 q^{74} + 448 q^{76} - 672 q^{77} - 432 q^{79} + 1032 q^{80} + 2832 q^{82} + 1388 q^{83} + 1744 q^{85} - 840 q^{86} - 1524 q^{88} - 960 q^{89} - 2744 q^{92} + 572 q^{94} - 464 q^{95} - 532 q^{97} + 574 q^{98}+O(q^{100})$$ 2 * q - 2 * q^2 + 14 * q^4 - 24 * q^5 - 54 * q^8 - 32 * q^10 + 44 * q^11 - 26 * q^13 + 56 * q^14 - 30 * q^16 - 164 * q^17 + 48 * q^19 - 56 * q^20 - 380 * q^22 - 8 * q^23 + 150 * q^25 + 26 * q^26 - 112 * q^28 - 404 * q^29 + 40 * q^31 + 126 * q^32 + 276 * q^34 - 112 * q^35 - 100 * q^37 - 104 * q^38 + 592 * q^40 - 200 * q^41 - 616 * q^43 + 980 * q^44 + 1352 * q^46 + 324 * q^47 - 574 * q^49 + 1194 * q^50 - 182 * q^52 + 164 * q^53 + 144 * q^55 + 56 * q^56 - 268 * q^58 - 140 * q^59 + 628 * q^61 + 688 * q^62 - 194 * q^64 + 312 * q^65 - 472 * q^67 - 1372 * q^68 - 560 * q^70 - 428 * q^71 - 900 * q^73 - 684 * q^74 + 448 * q^76 - 672 * q^77 - 432 * q^79 + 1032 * q^80 + 2832 * q^82 + 1388 * q^83 + 1744 * q^85 - 840 * q^86 - 1524 * q^88 - 960 * q^89 - 2744 * q^92 + 572 * q^94 - 464 * q^95 - 532 * q^97 + 574 * 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
 −3.74166 3.74166
−4.74166 0 14.4833 −4.51669 0 −7.48331 −30.7417 0 21.4166
1.2 2.74166 0 −0.483315 −19.4833 0 7.48331 −23.2583 0 −53.4166
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$13$$ $$1$$

## 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 117.4.a.c 2
3.b odd 2 1 39.4.a.b 2
4.b odd 2 1 1872.4.a.t 2
12.b even 2 1 624.4.a.r 2
13.b even 2 1 1521.4.a.s 2
15.d odd 2 1 975.4.a.j 2
21.c even 2 1 1911.4.a.h 2
24.f even 2 1 2496.4.a.s 2
24.h odd 2 1 2496.4.a.bc 2
39.d odd 2 1 507.4.a.f 2
39.f even 4 2 507.4.b.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.a.b 2 3.b odd 2 1
117.4.a.c 2 1.a even 1 1 trivial
507.4.a.f 2 39.d odd 2 1
507.4.b.f 4 39.f even 4 2
624.4.a.r 2 12.b even 2 1
975.4.a.j 2 15.d odd 2 1
1521.4.a.s 2 13.b even 2 1
1872.4.a.t 2 4.b odd 2 1
1911.4.a.h 2 21.c even 2 1
2496.4.a.s 2 24.f even 2 1
2496.4.a.bc 2 24.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 2T_{2} - 13$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(117))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 2T - 13$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 24T + 88$$
$7$ $$T^{2} - 56$$
$11$ $$T^{2} - 44T - 1532$$
$13$ $$(T + 13)^{2}$$
$17$ $$T^{2} + 164T + 6500$$
$19$ $$T^{2} - 48T + 520$$
$23$ $$T^{2} + 8T - 32240$$
$29$ $$T^{2} + 404T + 32740$$
$31$ $$T^{2} - 40T - 9064$$
$37$ $$T^{2} + 100T - 8476$$
$41$ $$T^{2} + 200T - 113704$$
$43$ $$T^{2} + 616T + 57008$$
$47$ $$T^{2} - 324T + 11908$$
$53$ $$T^{2} - 164T - 194876$$
$59$ $$T^{2} + 140T - 17500$$
$61$ $$T^{2} - 628T - 160348$$
$67$ $$T^{2} + 472T - 348904$$
$71$ $$T^{2} + 428T - 52988$$
$73$ $$T^{2} + 900T + 121636$$
$79$ $$T^{2} + 432T - 61760$$
$83$ $$T^{2} - 1388 T + 424292$$
$89$ $$T^{2} + 960T - 275000$$
$97$ $$T^{2} + 532T - 606844$$