# Properties

 Label 342.3.d.a Level $342$ Weight $3$ Character orbit 342.d Analytic conductor $9.319$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$342 = 2 \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 342.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$9.31882504112$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ Defining polynomial: $$x^{2} + 2$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 38) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{-2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta q^{2} -2 q^{4} + q^{5} + 5 q^{7} + 2 \beta q^{8} +O(q^{10})$$ $$q -\beta q^{2} -2 q^{4} + q^{5} + 5 q^{7} + 2 \beta q^{8} -\beta q^{10} -5 q^{11} -12 \beta q^{13} -5 \beta q^{14} + 4 q^{16} + 25 q^{17} + 19 q^{19} -2 q^{20} + 5 \beta q^{22} + 10 q^{23} -24 q^{25} -24 q^{26} -10 q^{28} -30 \beta q^{29} -30 \beta q^{31} -4 \beta q^{32} -25 \beta q^{34} + 5 q^{35} -18 \beta q^{37} -19 \beta q^{38} + 2 \beta q^{40} + 30 \beta q^{41} + 5 q^{43} + 10 q^{44} -10 \beta q^{46} -5 q^{47} -24 q^{49} + 24 \beta q^{50} + 24 \beta q^{52} + 18 \beta q^{53} -5 q^{55} + 10 \beta q^{56} -60 q^{58} -60 \beta q^{59} + 95 q^{61} -60 q^{62} -8 q^{64} -12 \beta q^{65} + 78 \beta q^{67} -50 q^{68} -5 \beta q^{70} -25 q^{73} -36 q^{74} -38 q^{76} -25 q^{77} + 30 \beta q^{79} + 4 q^{80} + 60 q^{82} + 130 q^{83} + 25 q^{85} -5 \beta q^{86} -10 \beta q^{88} + 90 \beta q^{89} -60 \beta q^{91} -20 q^{92} + 5 \beta q^{94} + 19 q^{95} + 12 \beta q^{97} + 24 \beta q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 4q^{4} + 2q^{5} + 10q^{7} + O(q^{10})$$ $$2q - 4q^{4} + 2q^{5} + 10q^{7} - 10q^{11} + 8q^{16} + 50q^{17} + 38q^{19} - 4q^{20} + 20q^{23} - 48q^{25} - 48q^{26} - 20q^{28} + 10q^{35} + 10q^{43} + 20q^{44} - 10q^{47} - 48q^{49} - 10q^{55} - 120q^{58} + 190q^{61} - 120q^{62} - 16q^{64} - 100q^{68} - 50q^{73} - 72q^{74} - 76q^{76} - 50q^{77} + 8q^{80} + 120q^{82} + 260q^{83} + 50q^{85} - 40q^{92} + 38q^{95} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/342\mathbb{Z}\right)^\times$$.

 $$n$$ $$191$$ $$325$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
37.1
 1.41421i − 1.41421i
1.41421i 0 −2.00000 1.00000 0 5.00000 2.82843i 0 1.41421i
37.2 1.41421i 0 −2.00000 1.00000 0 5.00000 2.82843i 0 1.41421i
 $$n$$: 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
19.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 342.3.d.a 2
3.b odd 2 1 38.3.b.a 2
4.b odd 2 1 2736.3.o.h 2
12.b even 2 1 304.3.e.c 2
15.d odd 2 1 950.3.c.a 2
15.e even 4 2 950.3.d.a 4
19.b odd 2 1 inner 342.3.d.a 2
24.f even 2 1 1216.3.e.i 2
24.h odd 2 1 1216.3.e.j 2
57.d even 2 1 38.3.b.a 2
76.d even 2 1 2736.3.o.h 2
228.b odd 2 1 304.3.e.c 2
285.b even 2 1 950.3.c.a 2
285.j odd 4 2 950.3.d.a 4
456.l odd 2 1 1216.3.e.i 2
456.p even 2 1 1216.3.e.j 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.3.b.a 2 3.b odd 2 1
38.3.b.a 2 57.d even 2 1
304.3.e.c 2 12.b even 2 1
304.3.e.c 2 228.b odd 2 1
342.3.d.a 2 1.a even 1 1 trivial
342.3.d.a 2 19.b odd 2 1 inner
950.3.c.a 2 15.d odd 2 1
950.3.c.a 2 285.b even 2 1
950.3.d.a 4 15.e even 4 2
950.3.d.a 4 285.j odd 4 2
1216.3.e.i 2 24.f even 2 1
1216.3.e.i 2 456.l odd 2 1
1216.3.e.j 2 24.h odd 2 1
1216.3.e.j 2 456.p even 2 1
2736.3.o.h 2 4.b odd 2 1
2736.3.o.h 2 76.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} - 1$$ acting on $$S_{3}^{\mathrm{new}}(342, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$2 + T^{2}$$
$3$ $$T^{2}$$
$5$ $$( -1 + T )^{2}$$
$7$ $$( -5 + T )^{2}$$
$11$ $$( 5 + T )^{2}$$
$13$ $$288 + T^{2}$$
$17$ $$( -25 + T )^{2}$$
$19$ $$( -19 + T )^{2}$$
$23$ $$( -10 + T )^{2}$$
$29$ $$1800 + T^{2}$$
$31$ $$1800 + T^{2}$$
$37$ $$648 + T^{2}$$
$41$ $$1800 + T^{2}$$
$43$ $$( -5 + T )^{2}$$
$47$ $$( 5 + T )^{2}$$
$53$ $$648 + T^{2}$$
$59$ $$7200 + T^{2}$$
$61$ $$( -95 + T )^{2}$$
$67$ $$12168 + T^{2}$$
$71$ $$T^{2}$$
$73$ $$( 25 + T )^{2}$$
$79$ $$1800 + T^{2}$$
$83$ $$( -130 + T )^{2}$$
$89$ $$16200 + T^{2}$$
$97$ $$288 + T^{2}$$