Properties

 Label 368.2.a.a Level $368$ Weight $2$ Character orbit 368.a Self dual yes Analytic conductor $2.938$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$368 = 2^{4} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 368.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$2.93849479438$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 184) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

 $$f(q)$$ $$=$$ $$q - 3 q^{3} + 2 q^{7} + 6 q^{9} + O(q^{10})$$ $$q - 3 q^{3} + 2 q^{7} + 6 q^{9} - 5 q^{13} - 6 q^{17} - 6 q^{19} - 6 q^{21} - q^{23} - 5 q^{25} - 9 q^{27} + 9 q^{29} - 3 q^{31} - 8 q^{37} + 15 q^{39} + 3 q^{41} + 8 q^{43} - 7 q^{47} - 3 q^{49} + 18 q^{51} - 2 q^{53} + 18 q^{57} - 4 q^{59} - 10 q^{61} + 12 q^{63} - 8 q^{67} + 3 q^{69} - 7 q^{71} + 9 q^{73} + 15 q^{75} + 6 q^{79} + 9 q^{81} + 14 q^{83} - 27 q^{87} + 16 q^{89} - 10 q^{91} + 9 q^{93} + 6 q^{97} + O(q^{100})$$

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 −3.00000 0 0 0 2.00000 0 6.00000 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
$$2$$ $$1$$
$$23$$ $$1$$

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 368.2.a.a 1
3.b odd 2 1 3312.2.a.i 1
4.b odd 2 1 184.2.a.d 1
5.b even 2 1 9200.2.a.bj 1
8.b even 2 1 1472.2.a.m 1
8.d odd 2 1 1472.2.a.a 1
12.b even 2 1 1656.2.a.c 1
20.d odd 2 1 4600.2.a.a 1
20.e even 4 2 4600.2.e.a 2
23.b odd 2 1 8464.2.a.b 1
28.d even 2 1 9016.2.a.b 1
92.b even 2 1 4232.2.a.j 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
184.2.a.d 1 4.b odd 2 1
368.2.a.a 1 1.a even 1 1 trivial
1472.2.a.a 1 8.d odd 2 1
1472.2.a.m 1 8.b even 2 1
1656.2.a.c 1 12.b even 2 1
3312.2.a.i 1 3.b odd 2 1
4232.2.a.j 1 92.b even 2 1
4600.2.a.a 1 20.d odd 2 1
4600.2.e.a 2 20.e even 4 2
8464.2.a.b 1 23.b odd 2 1
9016.2.a.b 1 28.d even 2 1
9200.2.a.bj 1 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(368))$$:

 $$T_{3} + 3$$ $$T_{5}$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$3 + T$$
$5$ $$T$$
$7$ $$-2 + T$$
$11$ $$T$$
$13$ $$5 + T$$
$17$ $$6 + T$$
$19$ $$6 + T$$
$23$ $$1 + T$$
$29$ $$-9 + T$$
$31$ $$3 + T$$
$37$ $$8 + T$$
$41$ $$-3 + T$$
$43$ $$-8 + T$$
$47$ $$7 + T$$
$53$ $$2 + T$$
$59$ $$4 + T$$
$61$ $$10 + T$$
$67$ $$8 + T$$
$71$ $$7 + T$$
$73$ $$-9 + T$$
$79$ $$-6 + T$$
$83$ $$-14 + T$$
$89$ $$-16 + T$$
$97$ $$-6 + T$$