# Properties

 Label 192.7.e.b Level 192 Weight 7 Character orbit 192.e Self dual yes Analytic conductor 44.170 Analytic rank 0 Dimension 1 CM discriminant -3 Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$192 = 2^{6} \cdot 3$$ Weight: $$k$$ $$=$$ $$7$$ Character orbit: $$[\chi]$$ $$=$$ 192.e (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$44.1703840550$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 3) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 27q^{3} - 286q^{7} + 729q^{9} + O(q^{10})$$ $$q + 27q^{3} - 286q^{7} + 729q^{9} - 506q^{13} + 10582q^{19} - 7722q^{21} + 15625q^{25} + 19683q^{27} + 35282q^{31} + 89206q^{37} - 13662q^{39} - 111386q^{43} - 35853q^{49} + 285714q^{57} + 420838q^{61} - 208494q^{63} - 172874q^{67} + 638066q^{73} + 421875q^{75} - 204622q^{79} + 531441q^{81} + 144716q^{91} + 952614q^{93} - 56446q^{97} + O(q^{100})$$

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
65.1
 0
0 27.0000 0 0 0 −286.000 0 729.000 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 CM by $$\Q(\sqrt{-3})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 192.7.e.b 1
3.b odd 2 1 CM 192.7.e.b 1
4.b odd 2 1 192.7.e.a 1
8.b even 2 1 3.7.b.a 1
8.d odd 2 1 48.7.e.a 1
12.b even 2 1 192.7.e.a 1
24.f even 2 1 48.7.e.a 1
24.h odd 2 1 3.7.b.a 1
40.f even 2 1 75.7.c.a 1
40.i odd 4 2 75.7.d.a 2
56.h odd 2 1 147.7.b.a 1
72.j odd 6 2 81.7.d.a 2
72.n even 6 2 81.7.d.a 2
120.i odd 2 1 75.7.c.a 1
120.w even 4 2 75.7.d.a 2
168.i even 2 1 147.7.b.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.7.b.a 1 8.b even 2 1
3.7.b.a 1 24.h odd 2 1
48.7.e.a 1 8.d odd 2 1
48.7.e.a 1 24.f even 2 1
75.7.c.a 1 40.f even 2 1
75.7.c.a 1 120.i odd 2 1
75.7.d.a 2 40.i odd 4 2
75.7.d.a 2 120.w even 4 2
81.7.d.a 2 72.j odd 6 2
81.7.d.a 2 72.n even 6 2
147.7.b.a 1 56.h odd 2 1
147.7.b.a 1 168.i even 2 1
192.7.e.a 1 4.b odd 2 1
192.7.e.a 1 12.b even 2 1
192.7.e.b 1 1.a even 1 1 trivial
192.7.e.b 1 3.b odd 2 1 CM

## Hecke kernels

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

 $$T_{5}$$ $$T_{7} + 286$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - 27 T$$
$5$ $$( 1 - 125 T )( 1 + 125 T )$$
$7$ $$1 + 286 T + 117649 T^{2}$$
$11$ $$( 1 - 1331 T )( 1 + 1331 T )$$
$13$ $$1 + 506 T + 4826809 T^{2}$$
$17$ $$( 1 - 4913 T )( 1 + 4913 T )$$
$19$ $$1 - 10582 T + 47045881 T^{2}$$
$23$ $$( 1 - 12167 T )( 1 + 12167 T )$$
$29$ $$( 1 - 24389 T )( 1 + 24389 T )$$
$31$ $$1 - 35282 T + 887503681 T^{2}$$
$37$ $$1 - 89206 T + 2565726409 T^{2}$$
$41$ $$( 1 - 68921 T )( 1 + 68921 T )$$
$43$ $$1 + 111386 T + 6321363049 T^{2}$$
$47$ $$( 1 - 103823 T )( 1 + 103823 T )$$
$53$ $$( 1 - 148877 T )( 1 + 148877 T )$$
$59$ $$( 1 - 205379 T )( 1 + 205379 T )$$
$61$ $$1 - 420838 T + 51520374361 T^{2}$$
$67$ $$1 + 172874 T + 90458382169 T^{2}$$
$71$ $$( 1 - 357911 T )( 1 + 357911 T )$$
$73$ $$1 - 638066 T + 151334226289 T^{2}$$
$79$ $$1 + 204622 T + 243087455521 T^{2}$$
$83$ $$( 1 - 571787 T )( 1 + 571787 T )$$
$89$ $$( 1 - 704969 T )( 1 + 704969 T )$$
$97$ $$1 + 56446 T + 832972004929 T^{2}$$