# Properties

 Label 192.5.e.b Level 192 Weight 5 Character orbit 192.e Self dual yes Analytic conductor 19.847 Analytic rank 0 Dimension 1 CM discriminant -3 Inner twists 2

# Learn more about

## Newspace parameters

 Level: $$N$$ $$=$$ $$192 = 2^{6} \cdot 3$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 192.e (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$19.8470329121$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 12) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 9q^{3} + 94q^{7} + 81q^{9} + O(q^{10})$$ $$q + 9q^{3} + 94q^{7} + 81q^{9} - 146q^{13} - 46q^{19} + 846q^{21} + 625q^{25} + 729q^{27} - 194q^{31} + 2062q^{37} - 1314q^{39} - 3214q^{43} + 6435q^{49} - 414q^{57} + 1966q^{61} + 7614q^{63} + 5906q^{67} - 8542q^{73} + 5625q^{75} - 7682q^{79} + 6561q^{81} - 13724q^{91} - 1746q^{93} - 18814q^{97} + O(q^{100})$$

## 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 9.00000 0 0 0 94.0000 0 81.0000 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 192.5.e.b 1
3.b odd 2 1 CM 192.5.e.b 1
4.b odd 2 1 192.5.e.a 1
8.b even 2 1 48.5.e.a 1
8.d odd 2 1 12.5.c.a 1
12.b even 2 1 192.5.e.a 1
24.f even 2 1 12.5.c.a 1
24.h odd 2 1 48.5.e.a 1
40.e odd 2 1 300.5.g.b 1
40.k even 4 2 300.5.b.a 2
56.e even 2 1 588.5.c.a 1
72.l even 6 2 324.5.g.b 2
72.p odd 6 2 324.5.g.b 2
120.m even 2 1 300.5.g.b 1
120.q odd 4 2 300.5.b.a 2
168.e odd 2 1 588.5.c.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.5.c.a 1 8.d odd 2 1
12.5.c.a 1 24.f even 2 1
48.5.e.a 1 8.b even 2 1
48.5.e.a 1 24.h odd 2 1
192.5.e.a 1 4.b odd 2 1
192.5.e.a 1 12.b even 2 1
192.5.e.b 1 1.a even 1 1 trivial
192.5.e.b 1 3.b odd 2 1 CM
300.5.b.a 2 40.k even 4 2
300.5.b.a 2 120.q odd 4 2
300.5.g.b 1 40.e odd 2 1
300.5.g.b 1 120.m even 2 1
324.5.g.b 2 72.l even 6 2
324.5.g.b 2 72.p odd 6 2
588.5.c.a 1 56.e even 2 1
588.5.c.a 1 168.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 $$S_{5}^{\mathrm{new}}(192, [\chi])$$:

 $$T_{5}$$ $$T_{7} - 94$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - 9 T$$
$5$ $$( 1 - 25 T )( 1 + 25 T )$$
$7$ $$1 - 94 T + 2401 T^{2}$$
$11$ $$( 1 - 121 T )( 1 + 121 T )$$
$13$ $$1 + 146 T + 28561 T^{2}$$
$17$ $$( 1 - 289 T )( 1 + 289 T )$$
$19$ $$1 + 46 T + 130321 T^{2}$$
$23$ $$( 1 - 529 T )( 1 + 529 T )$$
$29$ $$( 1 - 841 T )( 1 + 841 T )$$
$31$ $$1 + 194 T + 923521 T^{2}$$
$37$ $$1 - 2062 T + 1874161 T^{2}$$
$41$ $$( 1 - 1681 T )( 1 + 1681 T )$$
$43$ $$1 + 3214 T + 3418801 T^{2}$$
$47$ $$( 1 - 2209 T )( 1 + 2209 T )$$
$53$ $$( 1 - 2809 T )( 1 + 2809 T )$$
$59$ $$( 1 - 3481 T )( 1 + 3481 T )$$
$61$ $$1 - 1966 T + 13845841 T^{2}$$
$67$ $$1 - 5906 T + 20151121 T^{2}$$
$71$ $$( 1 - 5041 T )( 1 + 5041 T )$$
$73$ $$1 + 8542 T + 28398241 T^{2}$$
$79$ $$1 + 7682 T + 38950081 T^{2}$$
$83$ $$( 1 - 6889 T )( 1 + 6889 T )$$
$89$ $$( 1 - 7921 T )( 1 + 7921 T )$$
$97$ $$1 + 18814 T + 88529281 T^{2}$$
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