# Properties

 Label 117.10.a.c Level $117$ Weight $10$ Character orbit 117.a Self dual yes Analytic conductor $60.259$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [117,10,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, 10, names="a")

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

chi := DirichletCharacter("117.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$117 = 3^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$10$$ 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: $$60.2591928312$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\mathbb{Q}[x]/(x^{4} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 1602x^{2} + 1544x + 342272$$ x^4 - x^3 - 1602*x^2 + 1544*x + 342272 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 13) 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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_1 + 8) q^{2} + (6 \beta_{3} + \beta_{2} + 15 \beta_1 + 352) q^{4} + ( - \beta_{3} + 9 \beta_{2} + \cdots - 113) q^{5}+ \cdots + (110 \beta_{3} + 129 \beta_{2} + \cdots + 11272) q^{8}+O(q^{10})$$ q + (b1 + 8) * q^2 + (6*b3 + b2 + 15*b1 + 352) * q^4 + (-b3 + 9*b2 - 18*b1 - 113) * q^5 + (-3*b3 + 50*b2 + 126*b1 - 2841) * q^7 + (110*b3 + 129*b2 + 345*b1 + 11272) * q^8 $$q + (\beta_1 + 8) q^{2} + (6 \beta_{3} + \beta_{2} + 15 \beta_1 + 352) q^{4} + ( - \beta_{3} + 9 \beta_{2} + \cdots - 113) q^{5}+ \cdots + ( - 7134408 \beta_{3} + \cdots - 106094368) q^{98}+O(q^{100})$$ q + (b1 + 8) * q^2 + (6*b3 + b2 + 15*b1 + 352) * q^4 + (-b3 + 9*b2 - 18*b1 - 113) * q^5 + (-3*b3 + 50*b2 + 126*b1 - 2841) * q^7 + (110*b3 + 129*b2 + 345*b1 + 11272) * q^8 + (72*b3 + 128*b2 - 159*b1 - 16936) * q^10 + (-436*b3 - 258*b2 + 344*b1 + 10058) * q^11 - 28561 * q^13 + (1756*b3 + 978*b2 - 1351*b1 + 69312) * q^14 + (1578*b3 + 3915*b2 + 15111*b1 + 177480) * q^16 + (885*b3 + 2751*b2 + 8794*b1 - 22099) * q^17 + (-432*b3 + 1906*b2 - 7944*b1 + 54510) * q^19 + (2118*b3 - 1311*b2 - 2177*b1 - 217696) * q^20 + (-3096*b3 - 11276*b2 - 19566*b1 + 346688) * q^22 + (14740*b3 - 12324*b2 + 6400*b1 + 1059076) * q^23 + (-11127*b3 - 7821*b2 + 9090*b1 - 724530) * q^25 + (-28561*b1 - 228488) * q^26 + (12990*b3 + 18749*b2 + 123375*b1 + 974704) * q^28 + (-33728*b3 + 13524*b2 + 29032*b1 + 413158) * q^29 + (-23982*b3 - 29782*b2 + 57276*b1 - 2849824) * q^31 + (112646*b3 + 44781*b2 + 270249*b1 + 7269016) * q^32 + (107784*b3 + 72472*b2 + 140115*b1 + 6502440) * q^34 + (-40537*b3 - 52668*b2 + 88466*b1 + 3435467) * q^35 + (1317*b3 - 39901*b2 + 278562*b1 + 1089253) * q^37 + (-9544*b3 + 19452*b2 + 1750*b1 - 6291168) * q^38 + (-76146*b3 - 57423*b2 - 36951*b1 + 5662472) * q^40 + (-51198*b3 - 10134*b2 - 230852*b1 - 3394372) * q^41 + (-226419*b3 + 42452*b2 + 316254*b1 - 8364479) * q^43 + (-119684*b3 - 139974*b2 - 344962*b1 - 16506144) * q^44 + (-208080*b3 + 20408*b2 + 1850052*b1 + 17431840) * q^46 + (-91721*b3 + 263046*b2 + 637746*b1 + 849245) * q^47 + (178833*b3 - 274077*b2 - 275478*b1 + 5847892) * q^49 + (-101880*b3 - 309720*b2 - 1498164*b1 + 1454448) * q^50 + (-171366*b3 - 28561*b2 - 428415*b1 - 10053472) * q^52 + (-192670*b3 - 171078*b2 + 1241756*b1 + 42492560) * q^53 + (119034*b3 + 631408*b2 - 656748*b1 - 30155706) * q^55 + (216158*b3 + 167961*b2 + 3661385*b1 + 69418856) * q^56 + (444672*b3 - 267184*b2 - 1325826*b1 + 20211472) * q^58 + (65820*b3 + 773310*b2 - 1469912*b1 + 16198370) * q^59 + (682194*b3 + 7562*b2 - 1308180*b1 + 19419044) * q^61 + (-251984*b3 - 862512*b2 - 4460252*b1 + 25147744) * q^62 + (1709178*b3 + 874163*b2 + 9349767*b1 + 189475880) * q^64 + (28561*b3 - 257049*b2 + 514098*b1 + 3227393) * q^65 + (-1620456*b3 + 256850*b2 + 810648*b1 - 10091066) * q^67 + (1837010*b3 + 1760643*b2 + 11038445*b1 + 176184992) * q^68 + (-522564*b3 - 1508150*b2 + 617673*b1 + 102240320) * q^70 + (-1020287*b3 - 380430*b2 + 5253814*b1 - 64173629) * q^71 + (-691596*b3 + 305096*b2 - 6481200*b1 + 150396966) * q^73 + (873352*b3 - 418584*b2 + 2485059*b1 + 238425032) * q^74 + (620724*b3 - 776690*b2 - 2511174*b1 - 81251680) * q^76 + (1722958*b3 + 2013870*b2 - 9461340*b1 - 138553018) * q^77 + (2407128*b3 + 905092*b2 - 525888*b1 + 28529436) * q^79 + (-2454582*b3 - 1617669*b2 + 726327*b1 + 127708872) * q^80 + (-1587792*b3 - 1232432*b2 - 8449152*b1 - 216277824) * q^82 + (-234878*b3 - 1501794*b2 - 10966732*b1 + 22694868) * q^83 + (-1810317*b3 - 2388931*b2 + 3047382*b1 + 137056601) * q^85 + (2746564*b3 - 2542314*b2 - 19962285*b1 + 151785152) * q^86 + (-3284100*b3 + 993874*b2 - 18802446*b1 - 576369456) * q^88 + (-3984044*b3 - 2129748*b2 - 21923976*b1 + 294537074) * q^89 + (85683*b3 - 1428050*b2 - 3598686*b1 + 81141801) * q^91 + (3961592*b3 + 5198004*b2 + 14114812*b1 + 1048851264) * q^92 + (9087396*b3 + 3905038*b2 + 3652059*b1 + 461792512) * q^94 + (-2850458*b3 - 1013268*b2 + 444292*b1 + 319027834) * q^95 + (-4723764*b3 + 5763088*b2 - 3356064*b1 + 264294478) * q^97 + (-7134408*b3 - 2347536*b2 + 10979626*b1 - 106094368) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 33 q^{2} + 1429 q^{4} - 471 q^{5} - 11241 q^{7} + 45543 q^{8}+O(q^{10})$$ 4 * q + 33 * q^2 + 1429 * q^4 - 471 * q^5 - 11241 * q^7 + 45543 * q^8 $$4 q + 33 q^{2} + 1429 q^{4} - 471 q^{5} - 11241 q^{7} + 45543 q^{8} - 67831 q^{10} + 40140 q^{11} - 114244 q^{13} + 277653 q^{14} + 726609 q^{16} - 78717 q^{17} + 209664 q^{19} - 870843 q^{20} + 1364090 q^{22} + 4257444 q^{23} - 2900157 q^{25} - 942513 q^{26} + 4035181 q^{28} + 1647936 q^{29} - 11366002 q^{31} + 29458959 q^{32} + 26257659 q^{34} + 13789797 q^{35} + 4636891 q^{37} - 25172466 q^{38} + 22536791 q^{40} - 13859538 q^{41} - 33368081 q^{43} - 66489222 q^{44} + 71369332 q^{46} + 3943005 q^{47} + 23294923 q^{49} + 4217748 q^{50} - 40813669 q^{52} + 171019326 q^{53} - 121160538 q^{55} + 281552967 q^{56} + 79964734 q^{58} + 63389388 q^{59} + 77050190 q^{61} + 95878740 q^{62} + 768962465 q^{64} + 13452231 q^{65} - 41174072 q^{67} + 717615423 q^{68} + 409056389 q^{70} - 252460989 q^{71} + 594415068 q^{73} + 957058539 q^{74} - 326897170 q^{76} - 561950454 q^{77} + 115998984 q^{79} + 509107233 q^{80} - 875148240 q^{82} + 79577862 q^{83} + 549463469 q^{85} + 589924887 q^{86} - 2327564370 q^{88} + 1152240276 q^{89} + 321054201 q^{91} + 4213481460 q^{92} + 1859909503 q^{94} + 1273705170 q^{95} + 1049098084 q^{97} - 420532254 q^{98}+O(q^{100})$$ 4 * q + 33 * q^2 + 1429 * q^4 - 471 * q^5 - 11241 * q^7 + 45543 * q^8 - 67831 * q^10 + 40140 * q^11 - 114244 * q^13 + 277653 * q^14 + 726609 * q^16 - 78717 * q^17 + 209664 * q^19 - 870843 * q^20 + 1364090 * q^22 + 4257444 * q^23 - 2900157 * q^25 - 942513 * q^26 + 4035181 * q^28 + 1647936 * q^29 - 11366002 * q^31 + 29458959 * q^32 + 26257659 * q^34 + 13789797 * q^35 + 4636891 * q^37 - 25172466 * q^38 + 22536791 * q^40 - 13859538 * q^41 - 33368081 * q^43 - 66489222 * q^44 + 71369332 * q^46 + 3943005 * q^47 + 23294923 * q^49 + 4217748 * q^50 - 40813669 * q^52 + 171019326 * q^53 - 121160538 * q^55 + 281552967 * q^56 + 79964734 * q^58 + 63389388 * q^59 + 77050190 * q^61 + 95878740 * q^62 + 768962465 * q^64 + 13452231 * q^65 - 41174072 * q^67 + 717615423 * q^68 + 409056389 * q^70 - 252460989 * q^71 + 594415068 * q^73 + 957058539 * q^74 - 326897170 * q^76 - 561950454 * q^77 + 115998984 * q^79 + 509107233 * q^80 - 875148240 * q^82 + 79577862 * q^83 + 549463469 * q^85 + 589924887 * q^86 - 2327564370 * q^88 + 1152240276 * q^89 + 321054201 * q^91 + 4213481460 * q^92 + 1859909503 * q^94 + 1273705170 * q^95 + 1049098084 * q^97 - 420532254 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 1602x^{2} + 1544x + 342272$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( 3\nu^{3} + 17\nu^{2} - 3586\nu - 12856 ) / 332$$ (3*v^3 + 17*v^2 - 3586*v - 12856) / 332 $$\beta_{3}$$ $$=$$ $$( -\nu^{3} + 105\nu^{2} + 1306\nu - 84248 ) / 664$$ (-v^3 + 105*v^2 + 1306*v - 84248) / 664
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$6\beta_{3} + \beta_{2} - \beta _1 + 800$$ 6*b3 + b2 - b1 + 800 $$\nu^{3}$$ $$=$$ $$-34\beta_{3} + 105\beta_{2} + 1201\beta _1 - 248$$ -34*b3 + 105*b2 + 1201*b1 - 248

## 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
 −36.8028 −15.3567 16.5360 36.6235
−28.8028 0 317.603 258.914 0 −8862.28 5599.18 0 −7457.46
1.2 −7.35673 0 −457.879 1236.25 0 892.010 7135.13 0 −9094.74
1.3 24.5360 0 90.0171 −1814.98 0 −8707.31 −10353.8 0 −44532.3
1.4 44.6235 0 1479.26 −151.187 0 5436.58 43162.5 0 −6746.48
 $$n$$: 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 117.10.a.c 4
3.b odd 2 1 13.10.a.a 4
12.b even 2 1 208.10.a.g 4
15.d odd 2 1 325.10.a.a 4
39.d odd 2 1 169.10.a.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.10.a.a 4 3.b odd 2 1
117.10.a.c 4 1.a even 1 1 trivial
169.10.a.a 4 39.d odd 2 1
208.10.a.g 4 12.b even 2 1
325.10.a.a 4 15.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} - 33T_{2}^{3} - 1194T_{2}^{2} + 24936T_{2} + 232000$$ acting on $$S_{10}^{\mathrm{new}}(\Gamma_0(117))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 33 T^{3} + \cdots + 232000$$
$3$ $$T^{4}$$
$5$ $$T^{4} + \cdots + 87830562190$$
$7$ $$T^{4} + \cdots + 374218195104754$$
$11$ $$T^{4} + \cdots + 27\!\cdots\!36$$
$13$ $$(T + 28561)^{4}$$
$17$ $$T^{4} + \cdots + 25\!\cdots\!18$$
$19$ $$T^{4} + \cdots + 50\!\cdots\!08$$
$23$ $$T^{4} + \cdots - 17\!\cdots\!36$$
$29$ $$T^{4} + \cdots + 37\!\cdots\!32$$
$31$ $$T^{4} + \cdots - 58\!\cdots\!60$$
$37$ $$T^{4} + \cdots + 61\!\cdots\!62$$
$41$ $$T^{4} + \cdots + 15\!\cdots\!52$$
$43$ $$T^{4} + \cdots + 26\!\cdots\!40$$
$47$ $$T^{4} + \cdots + 10\!\cdots\!50$$
$53$ $$T^{4} + \cdots - 27\!\cdots\!48$$
$59$ $$T^{4} + \cdots - 26\!\cdots\!68$$
$61$ $$T^{4} + \cdots + 60\!\cdots\!72$$
$67$ $$T^{4} + \cdots + 19\!\cdots\!64$$
$71$ $$T^{4} + \cdots + 45\!\cdots\!54$$
$73$ $$T^{4} + \cdots - 47\!\cdots\!72$$
$79$ $$T^{4} + \cdots + 25\!\cdots\!64$$
$83$ $$T^{4} + \cdots - 12\!\cdots\!88$$
$89$ $$T^{4} + \cdots - 34\!\cdots\!00$$
$97$ $$T^{4} + \cdots + 21\!\cdots\!60$$