# Properties

 Label 192.7.e.f Level $192$ Weight $7$ Character orbit 192.e Analytic conductor $44.170$ 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] = [192,7,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, 7, names="a")

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

chi := DirichletCharacter("192.65");

S:= CuspForms(chi, 7);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$192 = 2^{6} \cdot 3$$ Weight: $$k$$ $$=$$ $$7$$ 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: $$44.1703840550$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 2$$ x^2 + 2 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}\cdot 3$$ Twist minimal: no (minimal twist has level 6) 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 $$\beta = 12\sqrt{-2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta + 21) q^{3} + 10 \beta q^{5} - 2 q^{7} + (42 \beta + 153) q^{9}+O(q^{10})$$ q + (b + 21) * q^3 + 10*b * q^5 - 2 * q^7 + (42*b + 153) * q^9 $$q + (\beta + 21) q^{3} + 10 \beta q^{5} - 2 q^{7} + (42 \beta + 153) q^{9} - 2 \beta q^{11} + 2950 q^{13} + (210 \beta - 2880) q^{15} + 264 \beta q^{17} + 5258 q^{19} + ( - 2 \beta - 42) q^{21} + 604 \beta q^{23} - 13175 q^{25} + (1035 \beta - 8883) q^{27} - 130 \beta q^{29} - 22898 q^{31} + ( - 42 \beta + 576) q^{33} - 20 \beta q^{35} - 34058 q^{37} + (2950 \beta + 61950) q^{39} - 988 \beta q^{41} - 6406 q^{43} + (1530 \beta - 120960) q^{45} - 10600 \beta q^{47} - 117645 q^{49} + (5544 \beta - 76032) q^{51} + 11346 \beta q^{53} + 5760 q^{55} + (5258 \beta + 110418) q^{57} - 19258 \beta q^{59} + 62566 q^{61} + ( - 84 \beta - 306) q^{63} + 29500 \beta q^{65} + 438698 q^{67} + (12684 \beta - 173952) q^{69} + 4020 \beta q^{71} - 730510 q^{73} + ( - 13175 \beta - 276675) q^{75} + 4 \beta q^{77} - 340562 q^{79} + (12852 \beta - 484623) q^{81} + 29242 \beta q^{83} - 760320 q^{85} + ( - 2730 \beta + 37440) q^{87} - 22788 \beta q^{89} - 5900 q^{91} + ( - 22898 \beta - 480858) q^{93} + 52580 \beta q^{95} - 281086 q^{97} + ( - 306 \beta + 24192) q^{99} +O(q^{100})$$ q + (b + 21) * q^3 + 10*b * q^5 - 2 * q^7 + (42*b + 153) * q^9 - 2*b * q^11 + 2950 * q^13 + (210*b - 2880) * q^15 + 264*b * q^17 + 5258 * q^19 + (-2*b - 42) * q^21 + 604*b * q^23 - 13175 * q^25 + (1035*b - 8883) * q^27 - 130*b * q^29 - 22898 * q^31 + (-42*b + 576) * q^33 - 20*b * q^35 - 34058 * q^37 + (2950*b + 61950) * q^39 - 988*b * q^41 - 6406 * q^43 + (1530*b - 120960) * q^45 - 10600*b * q^47 - 117645 * q^49 + (5544*b - 76032) * q^51 + 11346*b * q^53 + 5760 * q^55 + (5258*b + 110418) * q^57 - 19258*b * q^59 + 62566 * q^61 + (-84*b - 306) * q^63 + 29500*b * q^65 + 438698 * q^67 + (12684*b - 173952) * q^69 + 4020*b * q^71 - 730510 * q^73 + (-13175*b - 276675) * q^75 + 4*b * q^77 - 340562 * q^79 + (12852*b - 484623) * q^81 + 29242*b * q^83 - 760320 * q^85 + (-2730*b + 37440) * q^87 - 22788*b * q^89 - 5900 * q^91 + (-22898*b - 480858) * q^93 + 52580*b * q^95 - 281086 * q^97 + (-306*b + 24192) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 42 q^{3} - 4 q^{7} + 306 q^{9}+O(q^{10})$$ 2 * q + 42 * q^3 - 4 * q^7 + 306 * q^9 $$2 q + 42 q^{3} - 4 q^{7} + 306 q^{9} + 5900 q^{13} - 5760 q^{15} + 10516 q^{19} - 84 q^{21} - 26350 q^{25} - 17766 q^{27} - 45796 q^{31} + 1152 q^{33} - 68116 q^{37} + 123900 q^{39} - 12812 q^{43} - 241920 q^{45} - 235290 q^{49} - 152064 q^{51} + 11520 q^{55} + 220836 q^{57} + 125132 q^{61} - 612 q^{63} + 877396 q^{67} - 347904 q^{69} - 1461020 q^{73} - 553350 q^{75} - 681124 q^{79} - 969246 q^{81} - 1520640 q^{85} + 74880 q^{87} - 11800 q^{91} - 961716 q^{93} - 562172 q^{97} + 48384 q^{99}+O(q^{100})$$ 2 * q + 42 * q^3 - 4 * q^7 + 306 * q^9 + 5900 * q^13 - 5760 * q^15 + 10516 * q^19 - 84 * q^21 - 26350 * q^25 - 17766 * q^27 - 45796 * q^31 + 1152 * q^33 - 68116 * q^37 + 123900 * q^39 - 12812 * q^43 - 241920 * q^45 - 235290 * q^49 - 152064 * q^51 + 11520 * q^55 + 220836 * q^57 + 125132 * q^61 - 612 * q^63 + 877396 * q^67 - 347904 * q^69 - 1461020 * q^73 - 553350 * q^75 - 681124 * q^79 - 969246 * q^81 - 1520640 * q^85 + 74880 * q^87 - 11800 * q^91 - 961716 * q^93 - 562172 * q^97 + 48384 * 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
 − 1.41421i 1.41421i
0 21.0000 16.9706i 0 169.706i 0 −2.00000 0 153.000 712.764i 0
65.2 0 21.0000 + 16.9706i 0 169.706i 0 −2.00000 0 153.000 + 712.764i 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 192.7.e.f 2
3.b odd 2 1 inner 192.7.e.f 2
4.b odd 2 1 192.7.e.c 2
8.b even 2 1 48.7.e.b 2
8.d odd 2 1 6.7.b.a 2
12.b even 2 1 192.7.e.c 2
24.f even 2 1 6.7.b.a 2
24.h odd 2 1 48.7.e.b 2
40.e odd 2 1 150.7.d.a 2
40.k even 4 2 150.7.b.a 4
56.e even 2 1 294.7.b.a 2
72.l even 6 2 162.7.d.b 4
72.p odd 6 2 162.7.d.b 4
120.m even 2 1 150.7.d.a 2
120.q odd 4 2 150.7.b.a 4
168.e odd 2 1 294.7.b.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.7.b.a 2 8.d odd 2 1
6.7.b.a 2 24.f even 2 1
48.7.e.b 2 8.b even 2 1
48.7.e.b 2 24.h odd 2 1
150.7.b.a 4 40.k even 4 2
150.7.b.a 4 120.q odd 4 2
150.7.d.a 2 40.e odd 2 1
150.7.d.a 2 120.m even 2 1
162.7.d.b 4 72.l even 6 2
162.7.d.b 4 72.p odd 6 2
192.7.e.c 2 4.b odd 2 1
192.7.e.c 2 12.b even 2 1
192.7.e.f 2 1.a even 1 1 trivial
192.7.e.f 2 3.b odd 2 1 inner
294.7.b.a 2 56.e even 2 1
294.7.b.a 2 168.e odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{7}^{\mathrm{new}}(192, [\chi])$$:

 $$T_{5}^{2} + 28800$$ T5^2 + 28800 $$T_{7} + 2$$ T7 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 42T + 729$$
$5$ $$T^{2} + 28800$$
$7$ $$(T + 2)^{2}$$
$11$ $$T^{2} + 1152$$
$13$ $$(T - 2950)^{2}$$
$17$ $$T^{2} + 20072448$$
$19$ $$(T - 5258)^{2}$$
$23$ $$T^{2} + 105067008$$
$29$ $$T^{2} + 4867200$$
$31$ $$(T + 22898)^{2}$$
$37$ $$(T + 34058)^{2}$$
$41$ $$T^{2} + 281129472$$
$43$ $$(T + 6406)^{2}$$
$47$ $$T^{2} + 32359680000$$
$53$ $$T^{2} + 37074734208$$
$59$ $$T^{2} + 106810722432$$
$61$ $$(T - 62566)^{2}$$
$67$ $$(T - 438698)^{2}$$
$71$ $$T^{2} + 4654195200$$
$73$ $$(T + 730510)^{2}$$
$79$ $$(T + 340562)^{2}$$
$83$ $$T^{2} + 246267234432$$
$89$ $$T^{2} + 149556367872$$
$97$ $$(T + 281086)^{2}$$