# Properties

 Label 208.8.i.b Level $208$ Weight $8$ Character orbit 208.i Analytic conductor $64.976$ Analytic rank $0$ Dimension $8$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [208,8,Mod(81,208)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(208, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2]))

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("208.81");

S:= CuspForms(chi, 8);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$208 = 2^{4} \cdot 13$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 208.i (of order $$3$$, degree $$2$$, not minimal)

## 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: no Analytic conductor: $$64.9760853007$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} + 4654x^{6} + 7012369x^{4} + 3763719168x^{2} + 637953638400$$ x^8 + 4654*x^6 + 7012369*x^4 + 3763719168*x^2 + 637953638400 Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$2^{6}\cdot 3$$ Twist minimal: no (minimal twist has level 26) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{3} q^{3} + ( - \beta_{4} + \beta_{3} - \beta_{2} + 69) q^{5} + (2 \beta_{6} + 7 \beta_{2} + \cdots + 138) q^{7}+ \cdots + ( - 6 \beta_{7} - 3 \beta_{6} + \cdots - 1302) q^{9}+O(q^{10})$$ q - b3 * q^3 + (-b4 + b3 - b2 + 69) * q^5 + (2*b6 + 7*b2 + 138*b1 + 138) * q^7 + (-6*b7 - 3*b6 + 6*b4 - 9*b2 - 1308*b1 - 1302) * q^9 $$q - \beta_{3} q^{3} + ( - \beta_{4} + \beta_{3} - \beta_{2} + 69) q^{5} + (2 \beta_{6} + 7 \beta_{2} + \cdots + 138) q^{7}+ \cdots + (1020 \beta_{5} + 7152 \beta_{4} + \cdots + 2592480) q^{99}+O(q^{100})$$ q - b3 * q^3 + (-b4 + b3 - b2 + 69) * q^5 + (2*b6 + 7*b2 + 138*b1 + 138) * q^7 + (-6*b7 - 3*b6 + 6*b4 - 9*b2 - 1308*b1 - 1302) * q^9 + (8*b6 + 8*b5 + 19*b3 - 1844*b1) * q^11 + (13*b7 + 13*b5 - 26*b3 + 13*b2 - 1092*b1 - 3783) * q^13 + (48*b7 - 12*b6 - 12*b5 - 206*b3 + 3900*b1) * q^15 + (34*b7 - 14*b6 - 34*b4 + 196*b2 + 7089*b1 + 7055) * q^17 + (56*b7 - 56*b4 + 259*b2 + 25000*b1 + 24944) * q^19 + (-87*b5 - 126*b4 - 547*b3 + 547*b2 + 22749) * q^21 + (-176*b7 + 74*b6 + 74*b5 + 399*b3 - 8398*b1) * q^23 + (-43*b5 - 168*b4 + 1561*b3 - 1561*b2 + 21657) * q^25 + (216*b5 - 72*b4 + 591*b3 - 591*b2 - 26712) * q^27 + (365*b7 + 187*b6 + 187*b5 + 600*b3 - 23009*b1) * q^29 + (88*b5 - 100*b4 - 204*b3 + 204*b2 - 77884) * q^31 + (450*b7 + 321*b6 - 450*b4 - 289*b2 + 60045*b1 + 59595) * q^33 + (64*b7 - 12*b6 - 64*b4 + 2918*b2 - 35360*b1 - 35424) * q^35 + (281*b7 + 3*b6 + 3*b5 + 868*b3 + 2551*b1) * q^37 + (-156*b7 + 546*b6 + 390*b5 + 624*b4 + 5915*b3 + 676*b2 - 106470*b1 - 50076) * q^39 + (-862*b7 - 70*b6 - 70*b5 + 3560*b3 - 21189*b1) * q^41 + (1080*b7 + 560*b6 - 1080*b4 - 5285*b2 + 143136*b1 + 142056) * q^43 + (-1569*b7 - 294*b6 + 1569*b4 + 4743*b2 - 576780*b1 - 575211) * q^45 + (-692*b5 + 1080*b4 + 8648*b3 - 8648*b2 + 144436) * q^47 + (462*b7 + 231*b6 + 231*b5 + 14049*b3 + 179796*b1) * q^49 + (-636*b5 + 840*b4 - 11361*b3 + 11361*b2 + 683220) * q^51 + (2368*b5 - 689*b4 + 4153*b3 - 4153*b2 + 307309) * q^53 + (-1708*b7 - 156*b6 - 156*b5 + 12214*b3 - 274060*b1) * q^55 + (-1617*b5 + 798*b4 - 35451*b3 + 35451*b2 + 883323) * q^57 + (5712*b7 + 1568*b6 - 5712*b4 + 12799*b2 - 54236*b1 - 59948) * q^59 + (-729*b7 + 2353*b6 + 729*b4 - 43060*b2 + 172233*b1 + 172962) * q^61 + (-1644*b7 - 2028*b6 - 2028*b5 - 45810*b3 - 1489764*b1) * q^63 + (182*b7 + 559*b6 - 221*b5 + 6110*b4 - 14261*b3 - 4030*b2 + 1159210*b1 - 194220) * q^65 + (-1568*b7 - 2176*b6 - 2176*b5 + 8379*b3 + 815892*b1) * q^67 + (12894*b7 + 999*b6 - 12894*b4 - 17525*b2 + 1406955*b1 + 1394061) * q^69 + (-3940*b7 + 1038*b6 + 3940*b4 - 39029*b2 + 42302*b1 + 46242) * q^71 + (-2457*b5 + 1266*b4 - 40803*b3 + 40803*b2 + 1786334) * q^73 + (14616*b7 + 744*b6 + 744*b5 - 39575*b3 + 5557920*b1) * q^75 + (-4651*b5 + 714*b4 + 34501*b3 - 34501*b2 - 3476119) * q^77 + (3516*b5 - 9732*b4 + 35196*b3 - 35196*b2 + 1756864) * q^79 + (2520*b7 + 1260*b6 + 1260*b5 + 39204*b3 - 937611*b1) * q^81 + (-4644*b5 + 14520*b4 + 3948*b3 - 3948*b2 - 154740) * q^83 + (-2453*b7 + 4894*b6 + 2453*b4 - 21701*b2 + 2954970*b1 + 2957423) * q^85 + (-3876*b7 + 13446*b6 + 3876*b4 + 67667*b2 + 1800510*b1 + 1804386) * q^87 + (-9298*b7 - 9577*b6 - 9577*b5 - 116719*b3 + 2853621*b1) * q^89 + (4368*b7 - 2184*b6 - 4160*b5 - 3536*b4 + 23920*b3 - 103441*b2 - 5581784*b1 - 4849364) * q^91 + (6672*b7 + 792*b6 + 792*b5 + 76672*b3 - 742440*b1) * q^93 + (15368*b7 + 6188*b6 - 15368*b4 - 43606*b2 + 5844300*b1 + 5828932) * q^95 + (8330*b7 + 7973*b6 - 8330*b4 - 55489*b2 - 98849*b1 - 107179) * q^97 + (1020*b5 + 7152*b4 - 136674*b3 + 136674*b2 + 2592480) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 556 q^{5} + 548 q^{7} - 5214 q^{9}+O(q^{10})$$ 8 * q + 556 * q^5 + 548 * q^7 - 5214 * q^9 $$8 q + 556 q^{5} + 548 q^{7} - 5214 q^{9} + 7392 q^{11} - 25818 q^{13} - 15528 q^{15} + 28316 q^{17} + 99888 q^{19} + 182148 q^{21} + 33388 q^{23} + 173756 q^{25} - 212544 q^{27} + 93140 q^{29} - 622320 q^{31} + 238638 q^{33} - 141544 q^{35} - 9636 q^{37} + 22932 q^{39} + 82892 q^{41} + 569264 q^{43} - 2303394 q^{45} + 1148400 q^{47} - 717798 q^{49} + 5459856 q^{51} + 2470700 q^{53} + 1092512 q^{55} + 7056924 q^{57} - 231504 q^{59} + 685684 q^{61} + 5951712 q^{63} - 6216678 q^{65} - 3271056 q^{67} + 5600034 q^{69} + 175012 q^{71} + 14275780 q^{73} - 22200960 q^{75} - 27830412 q^{77} + 14107904 q^{79} + 3758004 q^{81} - 1314576 q^{83} + 11814998 q^{85} + 7182900 q^{87} - 11452234 q^{89} - 16457168 q^{91} + 2984688 q^{93} + 23334088 q^{95} - 428002 q^{97} + 20715312 q^{99}+O(q^{100})$$ 8 * q + 556 * q^5 + 548 * q^7 - 5214 * q^9 + 7392 * q^11 - 25818 * q^13 - 15528 * q^15 + 28316 * q^17 + 99888 * q^19 + 182148 * q^21 + 33388 * q^23 + 173756 * q^25 - 212544 * q^27 + 93140 * q^29 - 622320 * q^31 + 238638 * q^33 - 141544 * q^35 - 9636 * q^37 + 22932 * q^39 + 82892 * q^41 + 569264 * q^43 - 2303394 * q^45 + 1148400 * q^47 - 717798 * q^49 + 5459856 * q^51 + 2470700 * q^53 + 1092512 * q^55 + 7056924 * q^57 - 231504 * q^59 + 685684 * q^61 + 5951712 * q^63 - 6216678 * q^65 - 3271056 * q^67 + 5600034 * q^69 + 175012 * q^71 + 14275780 * q^73 - 22200960 * q^75 - 27830412 * q^77 + 14107904 * q^79 + 3758004 * q^81 - 1314576 * q^83 + 11814998 * q^85 + 7182900 * q^87 - 11452234 * q^89 - 16457168 * q^91 + 2984688 * q^93 + 23334088 * q^95 - 428002 * q^97 + 20715312 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 4654x^{6} + 7012369x^{4} + 3763719168x^{2} + 637953638400$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{7} + 4654\nu^{5} + 6213649\nu^{3} + 1905097728\nu - 3143761920 ) / 6287523840$$ (v^7 + 4654*v^5 + 6213649*v^3 + 1905097728*v - 3143761920) / 6287523840 $$\beta_{2}$$ $$=$$ $$( \nu^{4} + 2327\nu^{2} + 11808\nu + 798720 ) / 7872$$ (v^4 + 2327*v^2 + 11808*v + 798720) / 7872 $$\beta_{3}$$ $$=$$ $$( -\nu^{4} - 2327\nu^{2} + 11808\nu - 798720 ) / 7872$$ (-v^4 - 2327*v^2 + 11808*v - 798720) / 7872 $$\beta_{4}$$ $$=$$ $$( -\nu^{6} - 3766\nu^{4} - 3863881\nu^{2} - 854819328 ) / 661248$$ (-v^6 - 3766*v^4 - 3863881*v^2 - 854819328) / 661248 $$\beta_{5}$$ $$=$$ $$( \nu^{6} + 4018\nu^{4} + 4780909\nu^{2} + 1440612480 ) / 330624$$ (v^6 + 4018*v^4 + 4780909*v^2 + 1440612480) / 330624 $$\beta_{6}$$ $$=$$ $$( - 381 \nu^{7} - 13312 \nu^{6} - 1613430 \nu^{5} - 53487616 \nu^{4} - 2152864077 \nu^{3} + \cdots - 19177433333760 ) / 8802533376$$ (-381*v^7 - 13312*v^6 - 1613430*v^5 - 53487616*v^4 - 2152864077*v^3 - 63643460608*v^2 - 864056577024*v - 19177433333760) / 8802533376 $$\beta_{7}$$ $$=$$ $$( - 625 \nu^{7} - 6656 \nu^{6} - 2429518 \nu^{5} - 25066496 \nu^{4} - 2689763713 \nu^{3} + \cdots - 5685276180480 ) / 8802533376$$ (-625*v^7 - 6656*v^6 - 2429518*v^5 - 25066496*v^4 - 2689763713*v^3 - 25717991936*v^2 - 694817061888*v - 5685276180480) / 8802533376
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_{2} ) / 3$$ (b3 + b2) / 3 $$\nu^{2}$$ $$=$$ $$\beta_{5} + 2\beta_{4} + 3\beta_{3} - 3\beta_{2} - 1163$$ b5 + 2*b4 + 3*b3 - 3*b2 - 1163 $$\nu^{3}$$ $$=$$ $$( 48\beta_{7} - 144\beta_{6} - 72\beta_{5} - 24\beta_{4} - 1655\beta_{3} - 1655\beta_{2} - 17760\beta _1 - 8904 ) / 3$$ (48*b7 - 144*b6 - 72*b5 - 24*b4 - 1655*b3 - 1655*b2 - 17760*b1 - 8904) / 3 $$\nu^{4}$$ $$=$$ $$-2327\beta_{5} - 4654\beta_{4} - 10917\beta_{3} + 10917\beta_{2} + 1907581$$ -2327*b5 - 4654*b4 - 10917*b3 + 10917*b2 + 1907581 $$\nu^{5}$$ $$=$$ $$( - 64464 \beta_{7} + 358704 \beta_{6} + 179352 \beta_{5} + 32232 \beta_{4} + 3087889 \beta_{3} + \cdots + 34452312 ) / 3$$ (-64464*b7 + 358704*b6 + 179352*b5 + 32232*b4 + 3087889*b3 + 3087889*b2 + 68840160*b1 + 34452312) / 3 $$\nu^{6}$$ $$=$$ $$4899601\beta_{5} + 9137954\beta_{4} + 29521779\beta_{3} - 29521779\beta_{2} - 3545075771$$ 4899601*b5 + 9137954*b4 + 29521779*b3 - 29521779*b2 - 3545075771 $$\nu^{7}$$ $$=$$ $$( 1760304 \beta_{7} - 774642960 \beta_{6} - 387321480 \beta_{5} - 880152 \beta_{4} - 5992544039 \beta_{3} + \cdots - 95583443592 ) / 3$$ (1760304*b7 - 774642960*b6 - 387321480*b5 - 880152*b4 - 5992544039*b3 - 5992544039*b2 - 191165126880*b1 - 95583443592) / 3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/208\mathbb{Z}\right)^\times$$.

 $$n$$ $$53$$ $$79$$ $$145$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1 - \beta_{1}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
81.1
 41.0833i 23.0850i − 18.4011i − 45.7672i − 41.0833i − 23.0850i 18.4011i 45.7672i
0 −35.5792 61.6250i 0 523.489 0 −208.401 + 360.960i 0 −1438.26 + 2491.14i 0
81.2 0 −19.9922 34.6275i 0 −323.700 0 284.296 492.415i 0 294.123 509.436i 0
81.3 0 15.9358 + 27.6017i 0 −54.4265 0 −556.354 + 963.633i 0 585.599 1014.29i 0
81.4 0 39.6356 + 68.6509i 0 132.638 0 754.459 1306.76i 0 −2048.46 + 3548.04i 0
113.1 0 −35.5792 + 61.6250i 0 523.489 0 −208.401 360.960i 0 −1438.26 2491.14i 0
113.2 0 −19.9922 + 34.6275i 0 −323.700 0 284.296 + 492.415i 0 294.123 + 509.436i 0
113.3 0 15.9358 27.6017i 0 −54.4265 0 −556.354 963.633i 0 585.599 + 1014.29i 0
113.4 0 39.6356 68.6509i 0 132.638 0 754.459 + 1306.76i 0 −2048.46 3548.04i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 81.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 208.8.i.b 8
4.b odd 2 1 26.8.c.b 8
12.b even 2 1 234.8.h.b 8
13.c even 3 1 inner 208.8.i.b 8
52.i odd 6 1 338.8.a.j 4
52.j odd 6 1 26.8.c.b 8
52.j odd 6 1 338.8.a.i 4
52.l even 12 2 338.8.b.h 8
156.p even 6 1 234.8.h.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.8.c.b 8 4.b odd 2 1
26.8.c.b 8 52.j odd 6 1
208.8.i.b 8 1.a even 1 1 trivial
208.8.i.b 8 13.c even 3 1 inner
234.8.h.b 8 12.b even 2 1
234.8.h.b 8 156.p even 6 1
338.8.a.i 4 52.j odd 6 1
338.8.a.j 4 52.i odd 6 1
338.8.b.h 8 52.l even 12 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{8} + 6981 T_{3}^{6} + 70848 T_{3}^{5} + 41545881 T_{3}^{4} + 247294944 T_{3}^{3} + \cdots + 51674244710400$$ acting on $$S_{8}^{\mathrm{new}}(208, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8} + \cdots + 51674244710400$$
$5$ $$(T^{4} - 278 T^{3} + \cdots + 1223288100)^{2}$$
$7$ $$T^{8} + \cdots + 15\!\cdots\!00$$
$11$ $$T^{8} + \cdots + 28\!\cdots\!36$$
$13$ $$T^{8} + \cdots + 15\!\cdots\!21$$
$17$ $$T^{8} + \cdots + 35\!\cdots\!41$$
$19$ $$T^{8} + \cdots + 16\!\cdots\!00$$
$23$ $$T^{8} + \cdots + 64\!\cdots\!36$$
$29$ $$T^{8} + \cdots + 65\!\cdots\!69$$
$31$ $$(T^{4} + \cdots + 10\!\cdots\!28)^{2}$$
$37$ $$T^{8} + \cdots + 27\!\cdots\!41$$
$41$ $$T^{8} + \cdots + 75\!\cdots\!25$$
$43$ $$T^{8} + \cdots + 19\!\cdots\!96$$
$47$ $$(T^{4} + \cdots + 87\!\cdots\!28)^{2}$$
$53$ $$(T^{4} + \cdots + 10\!\cdots\!32)^{2}$$
$59$ $$T^{8} + \cdots + 87\!\cdots\!64$$
$61$ $$T^{8} + \cdots + 36\!\cdots\!25$$
$67$ $$T^{8} + \cdots + 16\!\cdots\!96$$
$71$ $$T^{8} + \cdots + 58\!\cdots\!44$$
$73$ $$(T^{4} + \cdots - 21\!\cdots\!40)^{2}$$
$79$ $$(T^{4} + \cdots + 46\!\cdots\!08)^{2}$$
$83$ $$(T^{4} + \cdots - 67\!\cdots\!00)^{2}$$
$89$ $$T^{8} + \cdots + 27\!\cdots\!36$$
$97$ $$T^{8} + \cdots + 23\!\cdots\!84$$