# Properties

 Label 342.2.g.b Level $342$ Weight $2$ Character orbit 342.g Analytic conductor $2.731$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

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## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$2.73088374913$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 38) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 + \zeta_{6} ) q^{2} -\zeta_{6} q^{4} -4 q^{7} + q^{8} +O(q^{10})$$ $$q + ( -1 + \zeta_{6} ) q^{2} -\zeta_{6} q^{4} -4 q^{7} + q^{8} -3 q^{11} -2 \zeta_{6} q^{13} + ( 4 - 4 \zeta_{6} ) q^{14} + ( -1 + \zeta_{6} ) q^{16} + ( -6 + 6 \zeta_{6} ) q^{17} + ( -2 - 3 \zeta_{6} ) q^{19} + ( 3 - 3 \zeta_{6} ) q^{22} -6 \zeta_{6} q^{23} + 5 \zeta_{6} q^{25} + 2 q^{26} + 4 \zeta_{6} q^{28} + 2 q^{31} -\zeta_{6} q^{32} -6 \zeta_{6} q^{34} -10 q^{37} + ( 5 - 2 \zeta_{6} ) q^{38} + ( 9 - 9 \zeta_{6} ) q^{41} + ( 4 - 4 \zeta_{6} ) q^{43} + 3 \zeta_{6} q^{44} + 6 q^{46} + 9 q^{49} -5 q^{50} + ( -2 + 2 \zeta_{6} ) q^{52} + 6 \zeta_{6} q^{53} -4 q^{56} + ( -9 + 9 \zeta_{6} ) q^{59} + 4 \zeta_{6} q^{61} + ( -2 + 2 \zeta_{6} ) q^{62} + q^{64} + 7 \zeta_{6} q^{67} + 6 q^{68} + ( -6 + 6 \zeta_{6} ) q^{71} + ( 1 - \zeta_{6} ) q^{73} + ( 10 - 10 \zeta_{6} ) q^{74} + ( -3 + 5 \zeta_{6} ) q^{76} + 12 q^{77} + ( 4 - 4 \zeta_{6} ) q^{79} + 9 \zeta_{6} q^{82} -3 q^{83} + 4 \zeta_{6} q^{86} -3 q^{88} + 6 \zeta_{6} q^{89} + 8 \zeta_{6} q^{91} + ( -6 + 6 \zeta_{6} ) q^{92} + ( -17 + 17 \zeta_{6} ) q^{97} + ( -9 + 9 \zeta_{6} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{2} - q^{4} - 8q^{7} + 2q^{8} + O(q^{10})$$ $$2q - q^{2} - q^{4} - 8q^{7} + 2q^{8} - 6q^{11} - 2q^{13} + 4q^{14} - q^{16} - 6q^{17} - 7q^{19} + 3q^{22} - 6q^{23} + 5q^{25} + 4q^{26} + 4q^{28} + 4q^{31} - q^{32} - 6q^{34} - 20q^{37} + 8q^{38} + 9q^{41} + 4q^{43} + 3q^{44} + 12q^{46} + 18q^{49} - 10q^{50} - 2q^{52} + 6q^{53} - 8q^{56} - 9q^{59} + 4q^{61} - 2q^{62} + 2q^{64} + 7q^{67} + 12q^{68} - 6q^{71} + q^{73} + 10q^{74} - q^{76} + 24q^{77} + 4q^{79} + 9q^{82} - 6q^{83} + 4q^{86} - 6q^{88} + 6q^{89} + 8q^{91} - 6q^{92} - 17q^{97} - 9q^{98} + O(q^{100})$$

## Character values

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

 $$n$$ $$191$$ $$325$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
163.1
 0.5 + 0.866025i 0.5 − 0.866025i
−0.500000 + 0.866025i 0 −0.500000 0.866025i 0 0 −4.00000 1.00000 0 0
235.1 −0.500000 0.866025i 0 −0.500000 + 0.866025i 0 0 −4.00000 1.00000 0 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
19.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 342.2.g.b 2
3.b odd 2 1 38.2.c.a 2
4.b odd 2 1 2736.2.s.m 2
12.b even 2 1 304.2.i.c 2
15.d odd 2 1 950.2.e.d 2
15.e even 4 2 950.2.j.e 4
19.c even 3 1 inner 342.2.g.b 2
19.c even 3 1 6498.2.a.s 1
19.d odd 6 1 6498.2.a.e 1
24.f even 2 1 1216.2.i.d 2
24.h odd 2 1 1216.2.i.h 2
57.d even 2 1 722.2.c.b 2
57.f even 6 1 722.2.a.d 1
57.f even 6 1 722.2.c.b 2
57.h odd 6 1 38.2.c.a 2
57.h odd 6 1 722.2.a.c 1
57.j even 18 6 722.2.e.i 6
57.l odd 18 6 722.2.e.j 6
76.g odd 6 1 2736.2.s.m 2
228.m even 6 1 304.2.i.c 2
228.m even 6 1 5776.2.a.g 1
228.n odd 6 1 5776.2.a.n 1
285.n odd 6 1 950.2.e.d 2
285.v even 12 2 950.2.j.e 4
456.u even 6 1 1216.2.i.d 2
456.x odd 6 1 1216.2.i.h 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.2.c.a 2 3.b odd 2 1
38.2.c.a 2 57.h odd 6 1
304.2.i.c 2 12.b even 2 1
304.2.i.c 2 228.m even 6 1
342.2.g.b 2 1.a even 1 1 trivial
342.2.g.b 2 19.c even 3 1 inner
722.2.a.c 1 57.h odd 6 1
722.2.a.d 1 57.f even 6 1
722.2.c.b 2 57.d even 2 1
722.2.c.b 2 57.f even 6 1
722.2.e.i 6 57.j even 18 6
722.2.e.j 6 57.l odd 18 6
950.2.e.d 2 15.d odd 2 1
950.2.e.d 2 285.n odd 6 1
950.2.j.e 4 15.e even 4 2
950.2.j.e 4 285.v even 12 2
1216.2.i.d 2 24.f even 2 1
1216.2.i.d 2 456.u even 6 1
1216.2.i.h 2 24.h odd 2 1
1216.2.i.h 2 456.x odd 6 1
2736.2.s.m 2 4.b odd 2 1
2736.2.s.m 2 76.g odd 6 1
5776.2.a.g 1 228.m even 6 1
5776.2.a.n 1 228.n odd 6 1
6498.2.a.e 1 19.d odd 6 1
6498.2.a.s 1 19.c even 3 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(342, [\chi])$$:

 $$T_{5}$$ $$T_{7} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$( 4 + T )^{2}$$
$11$ $$( 3 + T )^{2}$$
$13$ $$4 + 2 T + T^{2}$$
$17$ $$36 + 6 T + T^{2}$$
$19$ $$19 + 7 T + T^{2}$$
$23$ $$36 + 6 T + T^{2}$$
$29$ $$T^{2}$$
$31$ $$( -2 + T )^{2}$$
$37$ $$( 10 + T )^{2}$$
$41$ $$81 - 9 T + T^{2}$$
$43$ $$16 - 4 T + T^{2}$$
$47$ $$T^{2}$$
$53$ $$36 - 6 T + T^{2}$$
$59$ $$81 + 9 T + T^{2}$$
$61$ $$16 - 4 T + T^{2}$$
$67$ $$49 - 7 T + T^{2}$$
$71$ $$36 + 6 T + T^{2}$$
$73$ $$1 - T + T^{2}$$
$79$ $$16 - 4 T + T^{2}$$
$83$ $$( 3 + T )^{2}$$
$89$ $$36 - 6 T + T^{2}$$
$97$ $$289 + 17 T + T^{2}$$
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