# Properties

 Label 208.10.a.g Level $208$ Weight $10$ Character orbit 208.a Self dual yes Analytic conductor $107.127$ 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] = [208,10,Mod(1,208)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(208, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("208.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$208 = 2^{4} \cdot 13$$ Weight: $$k$$ $$=$$ $$10$$ Character orbit: $$[\chi]$$ $$=$$ 208.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: $$107.127453922$$ 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: $$2^{4}$$ 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.

 $$f(q)$$ $$=$$ $$q + (\beta_{3} + 41) q^{3} + ( - 6 \beta_{3} + 7 \beta_{2} + \cdots + 118) q^{5}+ \cdots + (137 \beta_{3} - 12 \beta_{2} + \cdots - 7457) q^{9}+O(q^{10})$$ q + (b3 + 41) * q^3 + (-6*b3 + 7*b2 - 2*b1 + 118) * q^5 + (-7*b3 + 10*b2 - 40*b1 + 2811) * q^7 + (137*b3 - 12*b2 - 45*b1 - 7457) * q^9 $$q + (\beta_{3} + 41) q^{3} + ( - 6 \beta_{3} + 7 \beta_{2} + \cdots + 118) q^{5}+ \cdots + (3676984 \beta_{3} - 535014 \beta_{2} + \cdots - 532259770) q^{99}+O(q^{100})$$ q + (b3 + 41) * q^3 + (-6*b3 + 7*b2 - 2*b1 + 118) * q^5 + (-7*b3 + 10*b2 - 40*b1 + 2811) * q^7 + (137*b3 - 12*b2 - 45*b1 - 7457) * q^9 + (-499*b3 + 63*b2 - 195*b1 + 9926) * q^11 - 28561 * q^13 + (101*b3 - 286*b2 + 222*b1 - 20873) * q^15 + (-940*b3 + 55*b2 - 2696*b1 + 19458) * q^17 + (-1622*b3 + 2054*b2 + 148*b1 - 52308) * q^19 + (7469*b3 - 1876*b2 - 2205*b1 + 286006) * q^21 + (4093*b3 + 10647*b2 - 1677*b1 + 1068046) * q^23 + (-13392*b3 + 2265*b2 - 5556*b1 - 727821) * q^25 + (-9038*b3 - 2919*b2 - 9279*b1 + 516461) * q^27 + (22163*b3 + 11565*b2 - 1959*b1 - 403552) * q^29 + (40037*b3 - 16055*b2 + 13727*b1 + 2847496) * q^31 + (-11353*b3 - 4137*b2 + 10341*b1 - 3607178) * q^33 + (-67795*b3 + 27258*b2 - 25410*b1 + 3437315) * q^35 + (-53783*b3 + 55100*b2 + 15199*b1 + 1159552) * q^37 + (-28561*b3 - 1171001) * q^39 + (14609*b3 + 36589*b2 + 46723*b1 + 3477684) * q^41 + (188120*b3 + 38299*b2 - 4153*b1 + 8398625) * q^43 + (66967*b3 - 118895*b2 + 46041*b1 - 4368964) * q^45 + (-16986*b3 - 74735*b2 + 188311*b1 + 962821) * q^47 + (31521*b3 + 147312*b2 - 126765*b1 + 5868439) * q^49 + (260605*b3 - 96064*b2 - 134976*b1 + 4957253) * q^51 + (385261*b3 - 192591*b2 - 21513*b1 - 42706664) * q^53 + (-487073*b3 + 368039*b2 - 263369*b1 + 30260376) * q^55 + (-133646*b3 - 56924*b2 + 107406*b1 - 12004900) * q^57 + (619759*b3 - 553939*b2 + 219371*b1 + 15863802) * q^59 + (678949*b3 + 3245*b2 + 10807*b1 + 19433096) * q^61 + (1325401*b3 - 297413*b2 + 283173*b1 + 39180866) * q^63 + (171366*b3 - 199927*b2 + 57122*b1 - 3370198) * q^65 + (1188248*b3 + 432208*b2 + 175358*b1 + 10698632) * q^67 + (2144683*b3 - 540399*b2 - 167103*b1 + 137061710) * q^69 + (-1612157*b3 + 591870*b2 + 211440*b1 - 63370319) * q^71 + (444001*b3 - 1135597*b2 - 830501*b1 + 148430868) * q^73 + (-1233696*b3 - 146580*b2 + 263124*b1 - 133674756) * q^75 + (-3481821*b3 + 1758863*b2 - 255007*b1 + 140056874) * q^77 + (-2323628*b3 - 83500*b2 - 988592*b1 - 29601528) * q^79 + (-2047075*b3 + 99531*b2 + 645003*b1 + 108964039) * q^81 + (443786*b3 - 678664*b2 - 2180458*b1 + 19835746) * q^83 + (-2933126*b3 + 1122809*b2 - 1266122*b1 + 136913288) * q^85 + (2483519*b3 - 804957*b2 - 987849*b1 + 272301268) * q^87 + (1312410*b3 + 2671634*b2 + 4801382*b1 - 287064058) * q^89 + (199927*b3 - 285610*b2 + 1142440*b1 - 80284971) * q^91 + (4293879*b3 + 697109*b2 - 1088343*b1 + 405814386) * q^93 + (-2700014*b3 - 150444*b2 - 1163712*b1 + 317713678) * q^95 + (-241771*b3 - 4481993*b2 + 1281095*b1 + 261093580) * q^97 + (3676984*b3 - 535014*b2 + 4981932*b1 - 532259770) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 163 q^{3} + 471 q^{5} + 11241 q^{7} - 29953 q^{9}+O(q^{10})$$ 4 * q + 163 * q^3 + 471 * q^5 + 11241 * q^7 - 29953 * q^9 $$4 q + 163 q^{3} + 471 q^{5} + 11241 q^{7} - 29953 q^{9} + 40140 q^{11} - 114244 q^{13} - 83307 q^{15} + 78717 q^{17} - 209664 q^{19} + 1138431 q^{21} + 4257444 q^{23} - 2900157 q^{25} + 2077801 q^{27} - 1647936 q^{29} + 11366002 q^{31} - 14413222 q^{33} + 13789797 q^{35} + 4636891 q^{37} - 4655443 q^{39} + 13859538 q^{41} + 33368081 q^{43} - 17423928 q^{45} + 3943005 q^{47} + 23294923 q^{49} + 19664471 q^{51} - 171019326 q^{53} + 121160538 q^{55} - 47829030 q^{57} + 63389388 q^{59} + 77050190 q^{61} + 155695476 q^{63} - 13452231 q^{65} + 41174072 q^{67} + 546642556 q^{69} - 252460989 q^{71} + 594415068 q^{73} - 533318748 q^{75} + 561950454 q^{77} - 115998984 q^{79} + 437803700 q^{81} + 79577862 q^{83} + 549463469 q^{85} + 1087526510 q^{87} - 1152240276 q^{89} - 321054201 q^{91} + 1618266556 q^{93} + 1273705170 q^{95} + 1049098084 q^{97} - 2132181050 q^{99}+O(q^{100})$$ 4 * q + 163 * q^3 + 471 * q^5 + 11241 * q^7 - 29953 * q^9 + 40140 * q^11 - 114244 * q^13 - 83307 * q^15 + 78717 * q^17 - 209664 * q^19 + 1138431 * q^21 + 4257444 * q^23 - 2900157 * q^25 + 2077801 * q^27 - 1647936 * q^29 + 11366002 * q^31 - 14413222 * q^33 + 13789797 * q^35 + 4636891 * q^37 - 4655443 * q^39 + 13859538 * q^41 + 33368081 * q^43 - 17423928 * q^45 + 3943005 * q^47 + 23294923 * q^49 + 19664471 * q^51 - 171019326 * q^53 + 121160538 * q^55 - 47829030 * q^57 + 63389388 * q^59 + 77050190 * q^61 + 155695476 * q^63 - 13452231 * q^65 + 41174072 * q^67 + 546642556 * q^69 - 252460989 * q^71 + 594415068 * q^73 - 533318748 * q^75 + 561950454 * q^77 - 115998984 * q^79 + 437803700 * q^81 + 79577862 * q^83 + 549463469 * q^85 + 1087526510 * q^87 - 1152240276 * q^89 - 321054201 * q^91 + 1618266556 * q^93 + 1273705170 * q^95 + 1049098084 * q^97 - 2132181050 * q^99

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

 $$\beta_{1}$$ $$=$$ $$( 3\nu^{3} + 17\nu^{2} - 2258\nu - 13188 ) / 332$$ (3*v^3 + 17*v^2 - 2258*v - 13188) / 332 $$\beta_{2}$$ $$=$$ $$( -7\nu^{3} + 71\nu^{2} + 9806\nu - 59200 ) / 664$$ (-7*v^3 + 71*v^2 + 9806*v - 59200) / 664 $$\beta_{3}$$ $$=$$ $$( -\nu^{3} + 105\nu^{2} - 22\nu - 84248 ) / 664$$ (-v^3 + 105*v^2 - 22*v - 84248) / 664
 $$\nu$$ $$=$$ $$( -\beta_{3} + \beta_{2} + \beta _1 + 2 ) / 8$$ (-b3 + b2 + b1 + 2) / 8 $$\nu^{2}$$ $$=$$ $$( 41\beta_{3} + 7\beta_{2} + 15\beta _1 + 6422 ) / 8$$ (41*b3 + 7*b2 + 15*b1 + 6422) / 8 $$\nu^{3}$$ $$=$$ $$( -985\beta_{3} + 713\beta_{2} + 1553\beta _1 + 282 ) / 8$$ (-985*b3 + 713*b2 + 1553*b1 + 282) / 8

## 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
 16.5360 −15.3567 36.6235 −36.8028
0 −49.9972 0 1814.98 0 8707.31 0 −17183.3 0
1.2 0 −42.6243 0 −1236.25 0 −892.010 0 −17866.2 0
1.3 0 51.0278 0 151.187 0 −5436.58 0 −17079.2 0
1.4 0 204.594 0 −258.914 0 8862.28 0 22175.6 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
$$2$$ $$-1$$
$$13$$ $$+1$$

## 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 208.10.a.g 4
4.b odd 2 1 13.10.a.a 4
12.b even 2 1 117.10.a.c 4
20.d odd 2 1 325.10.a.a 4
52.b 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 4.b odd 2 1
117.10.a.c 4 12.b even 2 1
169.10.a.a 4 52.b odd 2 1
208.10.a.g 4 1.a even 1 1 trivial
325.10.a.a 4 20.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} - 163T_{3}^{3} - 11105T_{3}^{2} + 422211T_{3} + 22248576$$ acting on $$S_{10}^{\mathrm{new}}(\Gamma_0(208))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} - 163 T^{3} + \cdots + 22248576$$
$5$ $$T^{4} + \cdots + 87830562190$$
$7$ $$T^{4} + \cdots + 374218195104754$$
$11$ $$T^{4} + \cdots + 27\!\cdots\!36$$
$13$ $$(T + 28561)^{4}$$
$17$ $$T^{4} + \cdots + 25\!\cdots\!18$$
$19$ $$T^{4} + \cdots + 50\!\cdots\!08$$
$23$ $$T^{4} + \cdots - 17\!\cdots\!36$$
$29$ $$T^{4} + \cdots + 37\!\cdots\!32$$
$31$ $$T^{4} + \cdots - 58\!\cdots\!60$$
$37$ $$T^{4} + \cdots + 61\!\cdots\!62$$
$41$ $$T^{4} + \cdots + 15\!\cdots\!52$$
$43$ $$T^{4} + \cdots + 26\!\cdots\!40$$
$47$ $$T^{4} + \cdots + 10\!\cdots\!50$$
$53$ $$T^{4} + \cdots - 27\!\cdots\!48$$
$59$ $$T^{4} + \cdots - 26\!\cdots\!68$$
$61$ $$T^{4} + \cdots + 60\!\cdots\!72$$
$67$ $$T^{4} + \cdots + 19\!\cdots\!64$$
$71$ $$T^{4} + \cdots + 45\!\cdots\!54$$
$73$ $$T^{4} + \cdots - 47\!\cdots\!72$$
$79$ $$T^{4} + \cdots + 25\!\cdots\!64$$
$83$ $$T^{4} + \cdots - 12\!\cdots\!88$$
$89$ $$T^{4} + \cdots - 34\!\cdots\!00$$
$97$ $$T^{4} + \cdots + 21\!\cdots\!60$$