# Properties

 Label 192.6.a.q Level $192$ Weight $6$ Character orbit 192.a Self dual yes Analytic conductor $30.794$ Analytic rank $1$ 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] = [192,6,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, 6, names="a")

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

chi := DirichletCharacter("192.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$192 = 2^{6} \cdot 3$$ Weight: $$k$$ $$=$$ $$6$$ 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: $$30.7936934041$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{31})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 31$$ x^2 - 31 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 $$\beta = 16\sqrt{31}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 9 q^{3} + (\beta - 18) q^{5} + ( - \beta + 60) q^{7} + 81 q^{9}+O(q^{10})$$ q - 9 * q^3 + (b - 18) * q^5 + (-b + 60) * q^7 + 81 * q^9 $$q - 9 q^{3} + (\beta - 18) q^{5} + ( - \beta + 60) q^{7} + 81 q^{9} + ( - 6 \beta - 100) q^{11} + ( - 6 \beta - 142) q^{13} + ( - 9 \beta + 162) q^{15} + (6 \beta + 1338) q^{17} + (10 \beta + 36) q^{19} + (9 \beta - 540) q^{21} + (30 \beta - 1920) q^{23} + ( - 36 \beta + 5135) q^{25} - 729 q^{27} + (19 \beta - 5106) q^{29} + ( - 5 \beta - 5244) q^{31} + (54 \beta + 900) q^{33} + (78 \beta - 9016) q^{35} + ( - 84 \beta - 6574) q^{37} + (54 \beta + 1278) q^{39} + (50 \beta + 2082) q^{41} + ( - 130 \beta - 2916) q^{43} + (81 \beta - 1458) q^{45} + ( - 198 \beta - 760) q^{47} + ( - 120 \beta - 5271) q^{49} + ( - 54 \beta - 12042) q^{51} + (71 \beta - 4506) q^{53} + (8 \beta - 45816) q^{55} + ( - 90 \beta - 324) q^{57} + ( - 216 \beta - 27548) q^{59} + ( - 48 \beta + 31722) q^{61} + ( - 81 \beta + 4860) q^{63} + ( - 34 \beta - 45060) q^{65} + (488 \beta + 18396) q^{67} + ( - 270 \beta + 17280) q^{69} + (438 \beta - 18832) q^{71} + (552 \beta - 18918) q^{73} + (324 \beta - 46215) q^{75} + ( - 260 \beta + 41616) q^{77} + (295 \beta - 72444) q^{79} + 6561 q^{81} + ( - 222 \beta - 54636) q^{83} + (1230 \beta + 23532) q^{85} + ( - 171 \beta + 45954) q^{87} + ( - 572 \beta - 16278) q^{89} + ( - 218 \beta + 39096) q^{91} + (45 \beta + 47196) q^{93} + ( - 144 \beta + 78712) q^{95} + ( - 60 \beta + 34546) q^{97} + ( - 486 \beta - 8100) q^{99}+O(q^{100})$$ q - 9 * q^3 + (b - 18) * q^5 + (-b + 60) * q^7 + 81 * q^9 + (-6*b - 100) * q^11 + (-6*b - 142) * q^13 + (-9*b + 162) * q^15 + (6*b + 1338) * q^17 + (10*b + 36) * q^19 + (9*b - 540) * q^21 + (30*b - 1920) * q^23 + (-36*b + 5135) * q^25 - 729 * q^27 + (19*b - 5106) * q^29 + (-5*b - 5244) * q^31 + (54*b + 900) * q^33 + (78*b - 9016) * q^35 + (-84*b - 6574) * q^37 + (54*b + 1278) * q^39 + (50*b + 2082) * q^41 + (-130*b - 2916) * q^43 + (81*b - 1458) * q^45 + (-198*b - 760) * q^47 + (-120*b - 5271) * q^49 + (-54*b - 12042) * q^51 + (71*b - 4506) * q^53 + (8*b - 45816) * q^55 + (-90*b - 324) * q^57 + (-216*b - 27548) * q^59 + (-48*b + 31722) * q^61 + (-81*b + 4860) * q^63 + (-34*b - 45060) * q^65 + (488*b + 18396) * q^67 + (-270*b + 17280) * q^69 + (438*b - 18832) * q^71 + (552*b - 18918) * q^73 + (324*b - 46215) * q^75 + (-260*b + 41616) * q^77 + (295*b - 72444) * q^79 + 6561 * q^81 + (-222*b - 54636) * q^83 + (1230*b + 23532) * q^85 + (-171*b + 45954) * q^87 + (-572*b - 16278) * q^89 + (-218*b + 39096) * q^91 + (45*b + 47196) * q^93 + (-144*b + 78712) * q^95 + (-60*b + 34546) * q^97 + (-486*b - 8100) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 18 q^{3} - 36 q^{5} + 120 q^{7} + 162 q^{9}+O(q^{10})$$ 2 * q - 18 * q^3 - 36 * q^5 + 120 * q^7 + 162 * q^9 $$2 q - 18 q^{3} - 36 q^{5} + 120 q^{7} + 162 q^{9} - 200 q^{11} - 284 q^{13} + 324 q^{15} + 2676 q^{17} + 72 q^{19} - 1080 q^{21} - 3840 q^{23} + 10270 q^{25} - 1458 q^{27} - 10212 q^{29} - 10488 q^{31} + 1800 q^{33} - 18032 q^{35} - 13148 q^{37} + 2556 q^{39} + 4164 q^{41} - 5832 q^{43} - 2916 q^{45} - 1520 q^{47} - 10542 q^{49} - 24084 q^{51} - 9012 q^{53} - 91632 q^{55} - 648 q^{57} - 55096 q^{59} + 63444 q^{61} + 9720 q^{63} - 90120 q^{65} + 36792 q^{67} + 34560 q^{69} - 37664 q^{71} - 37836 q^{73} - 92430 q^{75} + 83232 q^{77} - 144888 q^{79} + 13122 q^{81} - 109272 q^{83} + 47064 q^{85} + 91908 q^{87} - 32556 q^{89} + 78192 q^{91} + 94392 q^{93} + 157424 q^{95} + 69092 q^{97} - 16200 q^{99}+O(q^{100})$$ 2 * q - 18 * q^3 - 36 * q^5 + 120 * q^7 + 162 * q^9 - 200 * q^11 - 284 * q^13 + 324 * q^15 + 2676 * q^17 + 72 * q^19 - 1080 * q^21 - 3840 * q^23 + 10270 * q^25 - 1458 * q^27 - 10212 * q^29 - 10488 * q^31 + 1800 * q^33 - 18032 * q^35 - 13148 * q^37 + 2556 * q^39 + 4164 * q^41 - 5832 * q^43 - 2916 * q^45 - 1520 * q^47 - 10542 * q^49 - 24084 * q^51 - 9012 * q^53 - 91632 * q^55 - 648 * q^57 - 55096 * q^59 + 63444 * q^61 + 9720 * q^63 - 90120 * q^65 + 36792 * q^67 + 34560 * q^69 - 37664 * q^71 - 37836 * q^73 - 92430 * q^75 + 83232 * q^77 - 144888 * q^79 + 13122 * q^81 - 109272 * q^83 + 47064 * q^85 + 91908 * q^87 - 32556 * q^89 + 78192 * q^91 + 94392 * q^93 + 157424 * q^95 + 69092 * q^97 - 16200 * 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
 −5.56776 5.56776
0 −9.00000 0 −107.084 0 149.084 0 81.0000 0
1.2 0 −9.00000 0 71.0842 0 −29.0842 0 81.0000 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 192.6.a.q 2
3.b odd 2 1 576.6.a.bn 2
4.b odd 2 1 192.6.a.r 2
8.b even 2 1 96.6.a.h yes 2
8.d odd 2 1 96.6.a.g 2
12.b even 2 1 576.6.a.bm 2
16.e even 4 2 768.6.d.s 4
16.f odd 4 2 768.6.d.z 4
24.f even 2 1 288.6.a.n 2
24.h odd 2 1 288.6.a.o 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
96.6.a.g 2 8.d odd 2 1
96.6.a.h yes 2 8.b even 2 1
192.6.a.q 2 1.a even 1 1 trivial
192.6.a.r 2 4.b odd 2 1
288.6.a.n 2 24.f even 2 1
288.6.a.o 2 24.h odd 2 1
576.6.a.bm 2 12.b even 2 1
576.6.a.bn 2 3.b odd 2 1
768.6.d.s 4 16.e even 4 2
768.6.d.z 4 16.f odd 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(192))$$:

 $$T_{5}^{2} + 36T_{5} - 7612$$ T5^2 + 36*T5 - 7612 $$T_{7}^{2} - 120T_{7} - 4336$$ T7^2 - 120*T7 - 4336

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T + 9)^{2}$$
$5$ $$T^{2} + 36T - 7612$$
$7$ $$T^{2} - 120T - 4336$$
$11$ $$T^{2} + 200T - 275696$$
$13$ $$T^{2} + 284T - 265532$$
$17$ $$T^{2} - 2676 T + 1504548$$
$19$ $$T^{2} - 72T - 792304$$
$23$ $$T^{2} + 3840 T - 3456000$$
$29$ $$T^{2} + 10212 T + 23206340$$
$31$ $$T^{2} + 10488 T + 27301136$$
$37$ $$T^{2} + 13148 T - 12778940$$
$41$ $$T^{2} - 4164 T - 15505276$$
$43$ $$T^{2} + 5832 T - 125615344$$
$47$ $$T^{2} + 1520 T - 310545344$$
$53$ $$T^{2} + 9012 T - 19701340$$
$59$ $$T^{2} + 55096 T + 388630288$$
$61$ $$T^{2} - 63444 T + 988000740$$
$67$ $$T^{2} - 36792 T - 1551497968$$
$71$ $$T^{2} + 37664 T - 1167829760$$
$73$ $$T^{2} + 37836 T - 2060240220$$
$79$ $$T^{2} + 144888 T + 4557502736$$
$83$ $$T^{2} + 109272 T + 2593974672$$
$89$ $$T^{2} + 32556 T - 2331558940$$
$97$ $$T^{2} - 69092 T + 1164856516$$