# Properties

 Label 348.2.a.d Level $348$ Weight $2$ Character orbit 348.a Self dual yes Analytic conductor $2.779$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$348 = 2^{2} \cdot 3 \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 348.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$2.77879399034$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{3} + 2q^{5} + q^{7} + q^{9} + O(q^{10})$$ $$q + q^{3} + 2q^{5} + q^{7} + q^{9} + q^{11} - 3q^{13} + 2q^{15} - 3q^{17} + 2q^{19} + q^{21} + 8q^{23} - q^{25} + q^{27} + q^{29} - 8q^{31} + q^{33} + 2q^{35} - 3q^{39} + 2q^{41} + 2q^{45} + 5q^{47} - 6q^{49} - 3q^{51} - 2q^{53} + 2q^{55} + 2q^{57} - 6q^{59} - 12q^{61} + q^{63} - 6q^{65} + 3q^{67} + 8q^{69} + 4q^{71} - 16q^{73} - q^{75} + q^{77} - 2q^{79} + q^{81} - 6q^{83} - 6q^{85} + q^{87} + 3q^{89} - 3q^{91} - 8q^{93} + 4q^{95} - 6q^{97} + q^{99} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
0 1.00000 0 2.00000 0 1.00000 0 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$29$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 348.2.a.d 1
3.b odd 2 1 1044.2.a.d 1
4.b odd 2 1 1392.2.a.g 1
5.b even 2 1 8700.2.a.c 1
5.c odd 4 2 8700.2.g.i 2
8.b even 2 1 5568.2.a.e 1
8.d odd 2 1 5568.2.a.u 1
12.b even 2 1 4176.2.a.i 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
348.2.a.d 1 1.a even 1 1 trivial
1044.2.a.d 1 3.b odd 2 1
1392.2.a.g 1 4.b odd 2 1
4176.2.a.i 1 12.b even 2 1
5568.2.a.e 1 8.b even 2 1
5568.2.a.u 1 8.d odd 2 1
8700.2.a.c 1 5.b even 2 1
8700.2.g.i 2 5.c odd 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} - 2$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(348))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$-1 + T$$
$5$ $$-2 + T$$
$7$ $$-1 + T$$
$11$ $$-1 + T$$
$13$ $$3 + T$$
$17$ $$3 + T$$
$19$ $$-2 + T$$
$23$ $$-8 + T$$
$29$ $$-1 + T$$
$31$ $$8 + T$$
$37$ $$T$$
$41$ $$-2 + T$$
$43$ $$T$$
$47$ $$-5 + T$$
$53$ $$2 + T$$
$59$ $$6 + T$$
$61$ $$12 + T$$
$67$ $$-3 + T$$
$71$ $$-4 + T$$
$73$ $$16 + T$$
$79$ $$2 + T$$
$83$ $$6 + T$$
$89$ $$-3 + T$$
$97$ $$6 + T$$
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