# Properties

 Label 192.8.a.r Level $192$ Weight $8$ Character orbit 192.a Self dual yes Analytic conductor $59.978$ Analytic rank $0$ 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] = [192,8,Mod(1,192)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(192, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("192.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$192 = 2^{6} \cdot 3$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 192.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: $$59.9779248930$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{235})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 235$$ x^2 - 235 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 96) 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 $$\beta = 16\sqrt{235}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 27 q^{3} + (\beta - 90) q^{5} + (5 \beta + 516) q^{7} + 729 q^{9}+O(q^{10})$$ q - 27 * q^3 + (b - 90) * q^5 + (5*b + 516) * q^7 + 729 * q^9 $$q - 27 q^{3} + (\beta - 90) q^{5} + (5 \beta + 516) q^{7} + 729 q^{9} + (6 \beta - 1420) q^{11} + (42 \beta + 170) q^{13} + ( - 27 \beta + 2430) q^{15} + (78 \beta + 4890) q^{17} + (70 \beta - 16020) q^{19} + ( - 135 \beta - 13932) q^{21} + (330 \beta + 5568) q^{23} + ( - 180 \beta - 9865) q^{25} - 19683 q^{27} + ( - 5 \beta - 152106) q^{29} + ( - 383 \beta - 38820) q^{31} + ( - 162 \beta + 38340) q^{33} + (66 \beta + 254360) q^{35} + (12 \beta - 507910) q^{37} + ( - 1134 \beta - 4590) q^{39} + ( - 1030 \beta + 352050) q^{41} + ( - 1150 \beta + 197748) q^{43} + (729 \beta - 65610) q^{45} + (1710 \beta + 578744) q^{47} + (5160 \beta + 946713) q^{49} + ( - 2106 \beta - 132030) q^{51} + ( - 1897 \beta - 784290) q^{53} + ( - 1960 \beta + 488760) q^{55} + ( - 1890 \beta + 432540) q^{57} + (7128 \beta - 69620) q^{59} + (6960 \beta - 1301790) q^{61} + (3645 \beta + 376164) q^{63} + ( - 3610 \beta + 2511420) q^{65} + ( - 1960 \beta + 2644884) q^{67} + ( - 8910 \beta - 150336) q^{69} + ( - 5262 \beta + 2860880) q^{71} + ( - 9336 \beta - 595350) q^{73} + (4860 \beta + 266355) q^{75} + ( - 4004 \beta + 1072080) q^{77} + (16597 \beta - 199140) q^{79} + 531441 q^{81} + ( - 17490 \beta + 3493308) q^{83} + ( - 2130 \beta + 4252380) q^{85} + (135 \beta + 4106862) q^{87} + (22420 \beta - 4083366) q^{89} + (22522 \beta + 12721320) q^{91} + (10341 \beta + 1048140) q^{93} + ( - 22320 \beta + 5653000) q^{95} + (20148 \beta - 5180750) q^{97} + (4374 \beta - 1035180) q^{99}+O(q^{100})$$ q - 27 * q^3 + (b - 90) * q^5 + (5*b + 516) * q^7 + 729 * q^9 + (6*b - 1420) * q^11 + (42*b + 170) * q^13 + (-27*b + 2430) * q^15 + (78*b + 4890) * q^17 + (70*b - 16020) * q^19 + (-135*b - 13932) * q^21 + (330*b + 5568) * q^23 + (-180*b - 9865) * q^25 - 19683 * q^27 + (-5*b - 152106) * q^29 + (-383*b - 38820) * q^31 + (-162*b + 38340) * q^33 + (66*b + 254360) * q^35 + (12*b - 507910) * q^37 + (-1134*b - 4590) * q^39 + (-1030*b + 352050) * q^41 + (-1150*b + 197748) * q^43 + (729*b - 65610) * q^45 + (1710*b + 578744) * q^47 + (5160*b + 946713) * q^49 + (-2106*b - 132030) * q^51 + (-1897*b - 784290) * q^53 + (-1960*b + 488760) * q^55 + (-1890*b + 432540) * q^57 + (7128*b - 69620) * q^59 + (6960*b - 1301790) * q^61 + (3645*b + 376164) * q^63 + (-3610*b + 2511420) * q^65 + (-1960*b + 2644884) * q^67 + (-8910*b - 150336) * q^69 + (-5262*b + 2860880) * q^71 + (-9336*b - 595350) * q^73 + (4860*b + 266355) * q^75 + (-4004*b + 1072080) * q^77 + (16597*b - 199140) * q^79 + 531441 * q^81 + (-17490*b + 3493308) * q^83 + (-2130*b + 4252380) * q^85 + (135*b + 4106862) * q^87 + (22420*b - 4083366) * q^89 + (22522*b + 12721320) * q^91 + (10341*b + 1048140) * q^93 + (-22320*b + 5653000) * q^95 + (20148*b - 5180750) * q^97 + (4374*b - 1035180) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 54 q^{3} - 180 q^{5} + 1032 q^{7} + 1458 q^{9}+O(q^{10})$$ 2 * q - 54 * q^3 - 180 * q^5 + 1032 * q^7 + 1458 * q^9 $$2 q - 54 q^{3} - 180 q^{5} + 1032 q^{7} + 1458 q^{9} - 2840 q^{11} + 340 q^{13} + 4860 q^{15} + 9780 q^{17} - 32040 q^{19} - 27864 q^{21} + 11136 q^{23} - 19730 q^{25} - 39366 q^{27} - 304212 q^{29} - 77640 q^{31} + 76680 q^{33} + 508720 q^{35} - 1015820 q^{37} - 9180 q^{39} + 704100 q^{41} + 395496 q^{43} - 131220 q^{45} + 1157488 q^{47} + 1893426 q^{49} - 264060 q^{51} - 1568580 q^{53} + 977520 q^{55} + 865080 q^{57} - 139240 q^{59} - 2603580 q^{61} + 752328 q^{63} + 5022840 q^{65} + 5289768 q^{67} - 300672 q^{69} + 5721760 q^{71} - 1190700 q^{73} + 532710 q^{75} + 2144160 q^{77} - 398280 q^{79} + 1062882 q^{81} + 6986616 q^{83} + 8504760 q^{85} + 8213724 q^{87} - 8166732 q^{89} + 25442640 q^{91} + 2096280 q^{93} + 11306000 q^{95} - 10361500 q^{97} - 2070360 q^{99}+O(q^{100})$$ 2 * q - 54 * q^3 - 180 * q^5 + 1032 * q^7 + 1458 * q^9 - 2840 * q^11 + 340 * q^13 + 4860 * q^15 + 9780 * q^17 - 32040 * q^19 - 27864 * q^21 + 11136 * q^23 - 19730 * q^25 - 39366 * q^27 - 304212 * q^29 - 77640 * q^31 + 76680 * q^33 + 508720 * q^35 - 1015820 * q^37 - 9180 * q^39 + 704100 * q^41 + 395496 * q^43 - 131220 * q^45 + 1157488 * q^47 + 1893426 * q^49 - 264060 * q^51 - 1568580 * q^53 + 977520 * q^55 + 865080 * q^57 - 139240 * q^59 - 2603580 * q^61 + 752328 * q^63 + 5022840 * q^65 + 5289768 * q^67 - 300672 * q^69 + 5721760 * q^71 - 1190700 * q^73 + 532710 * q^75 + 2144160 * q^77 - 398280 * q^79 + 1062882 * q^81 + 6986616 * q^83 + 8504760 * q^85 + 8213724 * q^87 - 8166732 * q^89 + 25442640 * q^91 + 2096280 * q^93 + 11306000 * q^95 - 10361500 * q^97 - 2070360 * q^99

## 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
 −15.3297 15.3297
0 −27.0000 0 −335.275 0 −710.377 0 729.000 0
1.2 0 −27.0000 0 155.275 0 1742.38 0 729.000 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
$$2$$ $$1$$
$$3$$ $$1$$

## 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 192.8.a.r 2
3.b odd 2 1 576.8.a.bm 2
4.b odd 2 1 192.8.a.u 2
8.b even 2 1 96.8.a.g yes 2
8.d odd 2 1 96.8.a.d 2
12.b even 2 1 576.8.a.bl 2
24.f even 2 1 288.8.a.i 2
24.h odd 2 1 288.8.a.j 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
96.8.a.d 2 8.d odd 2 1
96.8.a.g yes 2 8.b even 2 1
192.8.a.r 2 1.a even 1 1 trivial
192.8.a.u 2 4.b odd 2 1
288.8.a.i 2 24.f even 2 1
288.8.a.j 2 24.h odd 2 1
576.8.a.bl 2 12.b even 2 1
576.8.a.bm 2 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{8}^{\mathrm{new}}(\Gamma_0(192))$$:

 $$T_{5}^{2} + 180T_{5} - 52060$$ T5^2 + 180*T5 - 52060 $$T_{7}^{2} - 1032T_{7} - 1237744$$ T7^2 - 1032*T7 - 1237744

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T + 27)^{2}$$
$5$ $$T^{2} + 180T - 52060$$
$7$ $$T^{2} - 1032 T - 1237744$$
$11$ $$T^{2} + 2840 T - 149360$$
$13$ $$T^{2} - 340 T - 106093340$$
$17$ $$T^{2} - 9780 T - 342101340$$
$19$ $$T^{2} + 32040 T - 38143600$$
$23$ $$T^{2} - 11136 T - 6520421376$$
$29$ $$T^{2} + 304212 T + 23134731236$$
$31$ $$T^{2} + 77640 T - 7317817840$$
$37$ $$T^{2} + 1015820 T + 257963905060$$
$41$ $$T^{2} - 704100 T + 60115458500$$
$43$ $$T^{2} - 395496 T - 40457328496$$
$47$ $$T^{2} - 1157488 T + 159030761536$$
$53$ $$T^{2} + 1568580 T + 398618486660$$
$59$ $$T^{2} + 139240 T - 3051785437040$$
$61$ $$T^{2} + 2603580 T - 1219589451900$$
$67$ $$T^{2} - 5289768 T + 6764300717456$$
$71$ $$T^{2} - 5721760 T + 6518885551360$$
$73$ $$T^{2} + 1190700 T - 4889157880860$$
$79$ $$T^{2} + 398280 T - 16532041465840$$
$83$ $$T^{2} - 6986616 T - 6199749233136$$
$89$ $$T^{2} + 8166732 T - 13565931134044$$
$97$ $$T^{2} + 10361500 T + 2418705617860$$