# Properties

 Label 192.9.e.g Level $192$ Weight $9$ Character orbit 192.e Analytic conductor $78.217$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [192,9,Mod(65,192)]

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, 1]))

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("192.65");

S:= CuspForms(chi, 9);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$192 = 2^{6} \cdot 3$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 192.e (of order $$2$$, degree $$1$$, 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: $$78.2166931317$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-110})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 110$$ x^2 + 110 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2\cdot 3$$ Twist minimal: no (minimal twist has level 12) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 $$\beta = 6\sqrt{-110}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta + 51) q^{3} + 18 \beta q^{5} - 3094 q^{7} + ( - 102 \beta - 1359) q^{9}+O(q^{10})$$ q + (-b + 51) * q^3 + 18*b * q^5 - 3094 * q^7 + (-102*b - 1359) * q^9 $$q + ( - \beta + 51) q^{3} + 18 \beta q^{5} - 3094 q^{7} + ( - 102 \beta - 1359) q^{9} + 18 \beta q^{11} + 7294 q^{13} + (918 \beta + 71280) q^{15} + 936 \beta q^{17} + 80326 q^{19} + (3094 \beta - 157794) q^{21} - 1548 \beta q^{23} - 892415 q^{25} + ( - 3843 \beta - 473229) q^{27} + 13734 \beta q^{29} + 435914 q^{31} + (918 \beta + 71280) q^{33} - 55692 \beta q^{35} - 1159298 q^{37} + ( - 7294 \beta + 371994) q^{39} - 43164 \beta q^{41} - 990266 q^{43} + ( - 24462 \beta + 7270560) q^{45} - 106488 \beta q^{47} + 3808035 q^{49} + (47736 \beta + 3706560) q^{51} - 160038 \beta q^{53} - 1283040 q^{55} + ( - 80326 \beta + 4096626) q^{57} - 25398 \beta q^{59} - 19369154 q^{61} + (315588 \beta + 4204746) q^{63} + 131292 \beta q^{65} + 28024294 q^{67} + ( - 78948 \beta - 6130080) q^{69} - 534852 \beta q^{71} - 25230142 q^{73} + (892415 \beta - 45513165) q^{75} - 55692 \beta q^{77} - 63401398 q^{79} + (277236 \beta - 39352959) q^{81} - 755514 \beta q^{83} - 66718080 q^{85} + (700434 \beta + 54386640) q^{87} - 1244196 \beta q^{89} - 22567636 q^{91} + ( - 435914 \beta + 22231614) q^{93} + 1445868 \beta q^{95} + 19550306 q^{97} + ( - 24462 \beta + 7270560) q^{99} +O(q^{100})$$ q + (-b + 51) * q^3 + 18*b * q^5 - 3094 * q^7 + (-102*b - 1359) * q^9 + 18*b * q^11 + 7294 * q^13 + (918*b + 71280) * q^15 + 936*b * q^17 + 80326 * q^19 + (3094*b - 157794) * q^21 - 1548*b * q^23 - 892415 * q^25 + (-3843*b - 473229) * q^27 + 13734*b * q^29 + 435914 * q^31 + (918*b + 71280) * q^33 - 55692*b * q^35 - 1159298 * q^37 + (-7294*b + 371994) * q^39 - 43164*b * q^41 - 990266 * q^43 + (-24462*b + 7270560) * q^45 - 106488*b * q^47 + 3808035 * q^49 + (47736*b + 3706560) * q^51 - 160038*b * q^53 - 1283040 * q^55 + (-80326*b + 4096626) * q^57 - 25398*b * q^59 - 19369154 * q^61 + (315588*b + 4204746) * q^63 + 131292*b * q^65 + 28024294 * q^67 + (-78948*b - 6130080) * q^69 - 534852*b * q^71 - 25230142 * q^73 + (892415*b - 45513165) * q^75 - 55692*b * q^77 - 63401398 * q^79 + (277236*b - 39352959) * q^81 - 755514*b * q^83 - 66718080 * q^85 + (700434*b + 54386640) * q^87 - 1244196*b * q^89 - 22567636 * q^91 + (-435914*b + 22231614) * q^93 + 1445868*b * q^95 + 19550306 * q^97 + (-24462*b + 7270560) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 102 q^{3} - 6188 q^{7} - 2718 q^{9}+O(q^{10})$$ 2 * q + 102 * q^3 - 6188 * q^7 - 2718 * q^9 $$2 q + 102 q^{3} - 6188 q^{7} - 2718 q^{9} + 14588 q^{13} + 142560 q^{15} + 160652 q^{19} - 315588 q^{21} - 1784830 q^{25} - 946458 q^{27} + 871828 q^{31} + 142560 q^{33} - 2318596 q^{37} + 743988 q^{39} - 1980532 q^{43} + 14541120 q^{45} + 7616070 q^{49} + 7413120 q^{51} - 2566080 q^{55} + 8193252 q^{57} - 38738308 q^{61} + 8409492 q^{63} + 56048588 q^{67} - 12260160 q^{69} - 50460284 q^{73} - 91026330 q^{75} - 126802796 q^{79} - 78705918 q^{81} - 133436160 q^{85} + 108773280 q^{87} - 45135272 q^{91} + 44463228 q^{93} + 39100612 q^{97} + 14541120 q^{99}+O(q^{100})$$ 2 * q + 102 * q^3 - 6188 * q^7 - 2718 * q^9 + 14588 * q^13 + 142560 * q^15 + 160652 * q^19 - 315588 * q^21 - 1784830 * q^25 - 946458 * q^27 + 871828 * q^31 + 142560 * q^33 - 2318596 * q^37 + 743988 * q^39 - 1980532 * q^43 + 14541120 * q^45 + 7616070 * q^49 + 7413120 * q^51 - 2566080 * q^55 + 8193252 * q^57 - 38738308 * q^61 + 8409492 * q^63 + 56048588 * q^67 - 12260160 * q^69 - 50460284 * q^73 - 91026330 * q^75 - 126802796 * q^79 - 78705918 * q^81 - 133436160 * q^85 + 108773280 * q^87 - 45135272 * q^91 + 44463228 * q^93 + 39100612 * q^97 + 14541120 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/192\mathbb{Z}\right)^\times$$.

 $$n$$ $$65$$ $$127$$ $$133$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
65.1
 10.4881i − 10.4881i
0 51.0000 62.9285i 0 1132.71i 0 −3094.00 0 −1359.00 6418.71i 0
65.2 0 51.0000 + 62.9285i 0 1132.71i 0 −3094.00 0 −1359.00 + 6418.71i 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 192.9.e.g 2
3.b odd 2 1 inner 192.9.e.g 2
4.b odd 2 1 192.9.e.d 2
8.b even 2 1 12.9.c.b 2
8.d odd 2 1 48.9.e.c 2
12.b even 2 1 192.9.e.d 2
24.f even 2 1 48.9.e.c 2
24.h odd 2 1 12.9.c.b 2
40.f even 2 1 300.9.g.d 2
40.i odd 4 2 300.9.b.c 4
72.j odd 6 2 324.9.g.f 4
72.n even 6 2 324.9.g.f 4
120.i odd 2 1 300.9.g.d 2
120.w even 4 2 300.9.b.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.9.c.b 2 8.b even 2 1
12.9.c.b 2 24.h odd 2 1
48.9.e.c 2 8.d odd 2 1
48.9.e.c 2 24.f even 2 1
192.9.e.d 2 4.b odd 2 1
192.9.e.d 2 12.b even 2 1
192.9.e.g 2 1.a even 1 1 trivial
192.9.e.g 2 3.b odd 2 1 inner
300.9.b.c 4 40.i odd 4 2
300.9.b.c 4 120.w even 4 2
300.9.g.d 2 40.f even 2 1
300.9.g.d 2 120.i odd 2 1
324.9.g.f 4 72.j odd 6 2
324.9.g.f 4 72.n even 6 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{9}^{\mathrm{new}}(192, [\chi])$$:

 $$T_{5}^{2} + 1283040$$ T5^2 + 1283040 $$T_{7} + 3094$$ T7 + 3094

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 102T + 6561$$
$5$ $$T^{2} + 1283040$$
$7$ $$(T + 3094)^{2}$$
$11$ $$T^{2} + 1283040$$
$13$ $$(T - 7294)^{2}$$
$17$ $$T^{2} + 3469340160$$
$19$ $$(T - 80326)^{2}$$
$23$ $$T^{2} + 9489363840$$
$29$ $$T^{2} + 746946113760$$
$31$ $$(T - 435914)^{2}$$
$37$ $$(T + 1159298)^{2}$$
$41$ $$T^{2} + 7377998348160$$
$43$ $$(T + 990266)^{2}$$
$47$ $$T^{2} + 44905188810240$$
$53$ $$T^{2} + 101424159318240$$
$59$ $$T^{2} + 2554431279840$$
$61$ $$(T + 19369154)^{2}$$
$67$ $$(T - 28024294)^{2}$$
$71$ $$T^{2} + 11\!\cdots\!40$$
$73$ $$(T + 25230142)^{2}$$
$79$ $$(T + 63401398)^{2}$$
$83$ $$T^{2} + 22\!\cdots\!60$$
$89$ $$T^{2} + 61\!\cdots\!60$$
$97$ $$(T - 19550306)^{2}$$