# Properties

 Label 169.10.a.a Level $169$ Weight $10$ Character orbit 169.a Self dual yes Analytic conductor $87.041$ Analytic rank $1$ 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] = [169,10,Mod(1,169)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(169, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 10, names="a")

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

chi := DirichletCharacter("169.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$169 = 13^{2}$$ Weight: $$k$$ $$=$$ $$10$$ Character orbit: $$[\chi]$$ $$=$$ 169.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: $$87.0410563117$$ Analytic rank: $$1$$ 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: $$\Z[a_1, a_2, a_3]$$ 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.

 $$f(q)$$ $$=$$ $$q + (\beta_1 + 8) q^{2} + ( - \beta_{3} + 2 \beta_1 - 41) q^{3} + (6 \beta_{3} + \beta_{2} + 15 \beta_1 + 352) q^{4} + ( - \beta_{3} + 9 \beta_{2} + \cdots - 113) q^{5}+ \cdots + (125 \beta_{3} - 33 \beta_{2} + \cdots - 7400) q^{9}+O(q^{10})$$ q + (b1 + 8) * q^2 + (-b3 + 2*b1 - 41) * q^3 + (6*b3 + b2 + 15*b1 + 352) * q^4 + (-b3 + 9*b2 - 18*b1 - 113) * q^5 + (12*b3 - 14*b2 - 91*b1 + 1152) * q^6 + (3*b3 - 50*b2 - 126*b1 + 2841) * q^7 + (110*b3 + 129*b2 + 345*b1 + 11272) * q^8 + (125*b3 - 33*b2 - 478*b1 - 7400) * q^9 $$q + (\beta_1 + 8) q^{2} + ( - \beta_{3} + 2 \beta_1 - 41) q^{3} + (6 \beta_{3} + \beta_{2} + 15 \beta_1 + 352) q^{4} + ( - \beta_{3} + 9 \beta_{2} + \cdots - 113) q^{5}+ \cdots + (3141970 \beta_{3} + 5516946 \beta_{2} + \cdots - 536706688) q^{99}+O(q^{100})$$ q + (b1 + 8) * q^2 + (-b3 + 2*b1 - 41) * q^3 + (6*b3 + b2 + 15*b1 + 352) * q^4 + (-b3 + 9*b2 - 18*b1 - 113) * q^5 + (12*b3 - 14*b2 - 91*b1 + 1152) * q^6 + (3*b3 - 50*b2 - 126*b1 + 2841) * q^7 + (110*b3 + 129*b2 + 345*b1 + 11272) * q^8 + (125*b3 - 33*b2 - 478*b1 - 7400) * q^9 + (72*b3 + 128*b2 - 159*b1 - 16936) * q^10 + (-436*b3 - 258*b2 + 344*b1 + 10058) * q^11 + (-314*b3 - 151*b2 + 35*b1 - 38800) * q^12 + (-1756*b3 - 978*b2 + 1351*b1 - 69312) * q^14 + (-185*b3 + 508*b2 + 114*b1 - 20809) * q^15 + (1578*b3 + 3915*b2 + 15111*b1 + 177480) * q^16 + (-885*b3 - 2751*b2 - 8794*b1 + 22099) * q^17 + (-3528*b3 + 928*b2 - 3274*b1 - 421056) * q^18 + (432*b3 - 1906*b2 + 7944*b1 - 54510) * q^19 + (2118*b3 - 1311*b2 - 2177*b1 - 217696) * q^20 + (-5593*b3 + 329*b2 + 27510*b1 - 290087) * q^21 + (-3096*b3 - 11276*b2 - 19566*b1 + 346688) * q^22 + (-14740*b3 + 12324*b2 - 6400*b1 - 1059076) * q^23 + (-8954*b3 - 539*b2 - 14475*b1 - 884536) * q^24 + (-11127*b3 - 7821*b2 + 9090*b1 - 724530) * q^25 + (11957*b3 + 6360*b2 + 24878*b1 - 528659) * q^27 + (-12990*b3 - 18749*b2 - 123375*b1 - 974704) * q^28 + (33728*b3 - 13524*b2 - 29032*b1 - 413158) * q^29 + (10844*b3 + 6298*b2 - 23723*b1 - 182816) * q^30 + (23982*b3 + 29782*b2 - 57276*b1 + 2849824) * q^31 + (112646*b3 + 44781*b2 + 270249*b1 + 7269016) * q^32 + (15490*b3 - 14478*b2 - 55796*b1 + 3613382) * q^33 + (-107784*b3 - 72472*b2 - 140115*b1 - 6502440) * q^34 + (40537*b3 + 52668*b2 - 88466*b1 - 3435467) * q^35 + (-65084*b3 - 26122*b2 - 410182*b1 - 2778112) * q^36 + (-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 + (171640*b3 - 56056*b2 - 450205*b1 + 18960872) * q^42 + (-226419*b3 + 42452*b2 + 316254*b1 - 8364479) * q^43 + (-119684*b3 - 139974*b2 - 344962*b1 - 16506144) * q^44 + (51928*b3 - 164936*b2 + 187560*b1 + 4296110) * q^45 + (208080*b3 - 20408*b2 - 1850052*b1 - 17431840) * q^46 + (-91721*b3 + 263046*b2 + 637746*b1 + 849245) * q^47 + (63138*b3 - 90129*b2 - 1585461*b1 + 225384) * q^48 + (178833*b3 - 274077*b2 - 275478*b1 + 5847892) * q^49 + (-101880*b3 - 309720*b2 - 1498164*b1 + 1454448) * q^50 + (-164541*b3 + 38912*b2 + 1253242*b1 - 5188293) * q^51 + (192670*b3 + 171078*b2 - 1241756*b1 - 42492560) * q^53 + (276468*b3 + 330670*b2 + 512495*b1 + 16039488) * q^54 + (119034*b3 + 631408*b2 - 656748*b1 - 30155706) * q^55 + (-216158*b3 - 167961*b2 - 3661385*b1 - 69418856) * q^56 + (190570*b3 - 164330*b2 - 583068*b1 + 12055382) * q^57 + (-444672*b3 + 267184*b2 + 1325826*b1 - 20211472) * q^58 + (65820*b3 + 773310*b2 - 1469912*b1 + 16198370) * q^59 + (78342*b3 + 3049*b2 + 387539*b1 - 9543504) * q^60 + (682194*b3 + 7562*b2 - 1308180*b1 + 19419044) * q^61 + (251984*b3 + 862512*b2 + 4460252*b1 - 25147744) * q^62 + (1027988*b3 + 580586*b2 - 2112936*b1 + 39195106) * q^63 + (1709178*b3 + 874163*b2 + 9349767*b1 + 189475880) * q^64 + (-624336*b3 - 68560*b2 + 3982522*b1 - 11438640) * q^66 + (1620456*b3 - 256850*b2 - 810648*b1 + 10091066) * q^67 + (-1837010*b3 - 1760643*b2 - 11038445*b1 - 176184992) * q^68 + (1604284*b3 + 373296*b2 - 6038576*b1 + 137769212) * q^69 + (522564*b3 + 1508150*b2 - 617673*b1 - 102240320) * q^70 + (-1020287*b3 - 380430*b2 + 5253814*b1 - 64173629) * q^71 + (-1177196*b3 - 2396858*b2 - 8556426*b1 - 138211408) * q^72 + (691596*b3 - 305096*b2 + 6481200*b1 - 150396966) * q^73 + (-873352*b3 + 418584*b2 - 2485059*b1 - 238425032) * q^74 + (1380276*b3 - 409704*b2 - 3226728*b1 + 133791300) * q^75 + (-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 + (-1947544*b3 + 545472*b2 + 6873224*b1 + 108219505) * q^81 + (-1587792*b3 - 1232432*b2 - 8449152*b1 - 216277824) * q^82 + (-234878*b3 - 1501794*b2 - 10966732*b1 + 22694868) * q^83 + (-958734*b3 + 1118579*b2 + 11812381*b1 - 29938272) * q^84 + (1810317*b3 + 2388931*b2 - 3047382*b1 - 137056601) * q^85 + (2746564*b3 - 2542314*b2 - 19962285*b1 + 151785152) * q^86 + (-1678562*b3 + 182892*b2 + 10528348*b1 - 274094074) * q^87 + (-3284100*b3 + 993874*b2 - 18802446*b1 - 576369456) * q^88 + (-3984044*b3 - 2129748*b2 - 21923976*b1 + 294537074) * q^89 + (-2173360*b3 - 1950440*b2 + 6293446*b1 + 218357488) * q^90 + (-3961592*b3 - 5198004*b2 - 14114812*b1 - 1048851264) * q^92 + (-4990988*b3 + 1785452*b2 + 11546912*b1 - 406205620) * q^93 + (9087396*b3 + 3905038*b2 + 3652059*b1 + 461792512) * q^94 + (2850458*b3 + 1013268*b2 - 444292*b1 - 319027834) * q^95 + (-6730898*b3 - 1921607*b2 - 862875*b1 - 790965064) * q^96 + (4723764*b3 - 5763088*b2 + 3356064*b1 - 264294478) * q^97 + (-7134408*b3 - 2347536*b2 + 10979626*b1 - 106094368) * q^98 + (3141970*b3 + 5516946*b2 + 11503732*b1 - 536706688) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 33 q^{2} - 163 q^{3} + 1429 q^{4} - 471 q^{5} + 4529 q^{6} + 11241 q^{7} + 45543 q^{8} - 29953 q^{9}+O(q^{10})$$ 4 * q + 33 * q^2 - 163 * q^3 + 1429 * q^4 - 471 * q^5 + 4529 * q^6 + 11241 * q^7 + 45543 * q^8 - 29953 * q^9 $$4 q + 33 q^{2} - 163 q^{3} + 1429 q^{4} - 471 q^{5} + 4529 q^{6} + 11241 q^{7} + 45543 q^{8} - 29953 q^{9} - 67831 q^{10} + 40140 q^{11} - 155479 q^{12} - 277653 q^{14} - 83307 q^{15} + 726609 q^{16} + 78717 q^{17} - 1691026 q^{18} - 209664 q^{19} - 870843 q^{20} - 1138431 q^{21} + 1364090 q^{22} - 4257444 q^{23} - 3561573 q^{24} - 2900157 q^{25} - 2077801 q^{27} - 4035181 q^{28} - 1647936 q^{29} - 744143 q^{30} + 11366002 q^{31} + 29458959 q^{32} + 14413222 q^{33} - 26257659 q^{34} - 13789797 q^{35} - 11587714 q^{36} - 4636891 q^{37} + 25172466 q^{38} + 22536791 q^{40} - 13859538 q^{41} + 75564923 q^{42} - 33368081 q^{43} - 66489222 q^{44} + 17423928 q^{45} - 71369332 q^{46} + 3943005 q^{47} - 620787 q^{48} + 23294923 q^{49} + 4217748 q^{50} - 19664471 q^{51} - 171019326 q^{53} + 64946915 q^{54} - 121160538 q^{55} - 281552967 q^{56} + 47829030 q^{57} - 79964734 q^{58} + 63389388 q^{59} - 37708135 q^{60} + 77050190 q^{61} - 95878740 q^{62} + 155695476 q^{63} + 768962465 q^{64} - 42396374 q^{66} + 41174072 q^{67} - 717615423 q^{68} + 546642556 q^{69} - 409056389 q^{70} - 252460989 q^{71} - 562579254 q^{72} - 594415068 q^{73} - 957058539 q^{74} + 533318748 q^{75} + 326897170 q^{76} + 561950454 q^{77} + 115998984 q^{79} + 509107233 q^{80} + 437803700 q^{81} - 875148240 q^{82} + 79577862 q^{83} - 108899441 q^{84} - 549463469 q^{85} + 589924887 q^{86} - 1087526510 q^{87} - 2327564370 q^{88} + 1152240276 q^{89} + 877550038 q^{90} - 4213481460 q^{92} - 1618266556 q^{93} + 1859909503 q^{94} - 1273705170 q^{95} - 3171454029 q^{96} - 1049098084 q^{97} - 420532254 q^{98} - 2132181050 q^{99}+O(q^{100})$$ 4 * q + 33 * q^2 - 163 * q^3 + 1429 * q^4 - 471 * q^5 + 4529 * q^6 + 11241 * q^7 + 45543 * q^8 - 29953 * q^9 - 67831 * q^10 + 40140 * q^11 - 155479 * q^12 - 277653 * q^14 - 83307 * q^15 + 726609 * q^16 + 78717 * q^17 - 1691026 * q^18 - 209664 * q^19 - 870843 * q^20 - 1138431 * q^21 + 1364090 * q^22 - 4257444 * q^23 - 3561573 * q^24 - 2900157 * q^25 - 2077801 * q^27 - 4035181 * q^28 - 1647936 * q^29 - 744143 * q^30 + 11366002 * q^31 + 29458959 * q^32 + 14413222 * q^33 - 26257659 * q^34 - 13789797 * q^35 - 11587714 * q^36 - 4636891 * q^37 + 25172466 * q^38 + 22536791 * q^40 - 13859538 * q^41 + 75564923 * q^42 - 33368081 * q^43 - 66489222 * q^44 + 17423928 * q^45 - 71369332 * q^46 + 3943005 * q^47 - 620787 * q^48 + 23294923 * q^49 + 4217748 * q^50 - 19664471 * q^51 - 171019326 * q^53 + 64946915 * q^54 - 121160538 * q^55 - 281552967 * q^56 + 47829030 * q^57 - 79964734 * q^58 + 63389388 * q^59 - 37708135 * q^60 + 77050190 * q^61 - 95878740 * q^62 + 155695476 * q^63 + 768962465 * q^64 - 42396374 * q^66 + 41174072 * q^67 - 717615423 * q^68 + 546642556 * q^69 - 409056389 * q^70 - 252460989 * q^71 - 562579254 * q^72 - 594415068 * q^73 - 957058539 * q^74 + 533318748 * q^75 + 326897170 * q^76 + 561950454 * q^77 + 115998984 * q^79 + 509107233 * q^80 + 437803700 * q^81 - 875148240 * q^82 + 79577862 * q^83 - 108899441 * q^84 - 549463469 * q^85 + 589924887 * q^86 - 1087526510 * q^87 - 2327564370 * q^88 + 1152240276 * q^89 + 877550038 * q^90 - 4213481460 * q^92 - 1618266556 * q^93 + 1859909503 * q^94 - 1273705170 * q^95 - 3171454029 * q^96 - 1049098084 * q^97 - 420532254 * q^98 - 2132181050 * q^99

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 −204.594 317.603 258.914 5892.88 8862.28 5599.18 22175.6 −7457.46
1.2 −7.35673 42.6243 −457.879 1236.25 −313.575 −892.010 7135.13 −17866.2 −9094.74
1.3 24.5360 49.9972 90.0171 −1814.98 1226.73 8707.31 −10353.8 −17183.3 −44532.3
1.4 44.6235 −51.0278 1479.26 −151.187 −2277.04 −5436.58 43162.5 −17079.2 −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
$$13$$ $$+1$$

## 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 169.10.a.a 4
13.b even 2 1 13.10.a.a 4
39.d odd 2 1 117.10.a.c 4
52.b odd 2 1 208.10.a.g 4
65.d even 2 1 325.10.a.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.10.a.a 4 13.b even 2 1
117.10.a.c 4 39.d odd 2 1
169.10.a.a 4 1.a even 1 1 trivial
208.10.a.g 4 52.b odd 2 1
325.10.a.a 4 65.d even 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 $$S_{10}^{\mathrm{new}}(\Gamma_0(169))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 33 T^{3} + \cdots + 232000$$
$3$ $$T^{4} + 163 T^{3} + \cdots + 22248576$$
$5$ $$T^{4} + \cdots + 87830562190$$
$7$ $$T^{4} + \cdots + 374218195104754$$
$11$ $$T^{4} + \cdots + 27\!\cdots\!36$$
$13$ $$T^{4}$$
$17$ $$T^{4} + \cdots + 25\!\cdots\!18$$
$19$ $$T^{4} + \cdots + 50\!\cdots\!08$$
$23$ $$T^{4} + \cdots - 17\!\cdots\!36$$
$29$ $$T^{4} + \cdots + 37\!\cdots\!32$$
$31$ $$T^{4} + \cdots - 58\!\cdots\!60$$
$37$ $$T^{4} + \cdots + 61\!\cdots\!62$$
$41$ $$T^{4} + \cdots + 15\!\cdots\!52$$
$43$ $$T^{4} + \cdots + 26\!\cdots\!40$$
$47$ $$T^{4} + \cdots + 10\!\cdots\!50$$
$53$ $$T^{4} + \cdots - 27\!\cdots\!48$$
$59$ $$T^{4} + \cdots - 26\!\cdots\!68$$
$61$ $$T^{4} + \cdots + 60\!\cdots\!72$$
$67$ $$T^{4} + \cdots + 19\!\cdots\!64$$
$71$ $$T^{4} + \cdots + 45\!\cdots\!54$$
$73$ $$T^{4} + \cdots - 47\!\cdots\!72$$
$79$ $$T^{4} + \cdots + 25\!\cdots\!64$$
$83$ $$T^{4} + \cdots - 12\!\cdots\!88$$
$89$ $$T^{4} + \cdots - 34\!\cdots\!00$$
$97$ $$T^{4} + \cdots + 21\!\cdots\!60$$