Properties

 Label 342.2.a.c Level $342$ Weight $2$ Character orbit 342.a Self dual yes Analytic conductor $2.731$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$342 = 2 \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 342.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$2.73088374913$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 114) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} - 4 q^{7} - q^{8}+O(q^{10})$$ q - q^2 + q^4 - 4 * q^7 - q^8 $$q - q^{2} + q^{4} - 4 q^{7} - q^{8} - 4 q^{13} + 4 q^{14} + q^{16} - 6 q^{17} + q^{19} + 6 q^{23} - 5 q^{25} + 4 q^{26} - 4 q^{28} - 6 q^{29} + 2 q^{31} - q^{32} + 6 q^{34} - 4 q^{37} - q^{38} - 6 q^{41} - 4 q^{43} - 6 q^{46} - 6 q^{47} + 9 q^{49} + 5 q^{50} - 4 q^{52} - 6 q^{53} + 4 q^{56} + 6 q^{58} + 12 q^{59} + 14 q^{61} - 2 q^{62} + q^{64} + 8 q^{67} - 6 q^{68} + 14 q^{73} + 4 q^{74} + q^{76} - 10 q^{79} + 6 q^{82} + 12 q^{83} + 4 q^{86} + 6 q^{89} + 16 q^{91} + 6 q^{92} + 6 q^{94} - 10 q^{97} - 9 q^{98}+O(q^{100})$$ q - q^2 + q^4 - 4 * q^7 - q^8 - 4 * q^13 + 4 * q^14 + q^16 - 6 * q^17 + q^19 + 6 * q^23 - 5 * q^25 + 4 * q^26 - 4 * q^28 - 6 * q^29 + 2 * q^31 - q^32 + 6 * q^34 - 4 * q^37 - q^38 - 6 * q^41 - 4 * q^43 - 6 * q^46 - 6 * q^47 + 9 * q^49 + 5 * q^50 - 4 * q^52 - 6 * q^53 + 4 * q^56 + 6 * q^58 + 12 * q^59 + 14 * q^61 - 2 * q^62 + q^64 + 8 * q^67 - 6 * q^68 + 14 * q^73 + 4 * q^74 + q^76 - 10 * q^79 + 6 * q^82 + 12 * q^83 + 4 * q^86 + 6 * q^89 + 16 * q^91 + 6 * q^92 + 6 * q^94 - 10 * q^97 - 9 * q^98

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
−1.00000 0 1.00000 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

Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$-1$$
$$19$$ $$-1$$

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 342.2.a.c 1
3.b odd 2 1 114.2.a.c 1
4.b odd 2 1 2736.2.a.o 1
5.b even 2 1 8550.2.a.bj 1
12.b even 2 1 912.2.a.c 1
15.d odd 2 1 2850.2.a.g 1
15.e even 4 2 2850.2.d.p 2
19.b odd 2 1 6498.2.a.t 1
21.c even 2 1 5586.2.a.u 1
24.f even 2 1 3648.2.a.bc 1
24.h odd 2 1 3648.2.a.i 1
57.d even 2 1 2166.2.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.2.a.c 1 3.b odd 2 1
342.2.a.c 1 1.a even 1 1 trivial
912.2.a.c 1 12.b even 2 1
2166.2.a.a 1 57.d even 2 1
2736.2.a.o 1 4.b odd 2 1
2850.2.a.g 1 15.d odd 2 1
2850.2.d.p 2 15.e even 4 2
3648.2.a.i 1 24.h odd 2 1
3648.2.a.bc 1 24.f even 2 1
5586.2.a.u 1 21.c even 2 1
6498.2.a.t 1 19.b odd 2 1
8550.2.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(342))$$:

 $$T_{5}$$ T5 $$T_{7} + 4$$ T7 + 4 $$T_{11}$$ T11

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T + 4$$
$11$ $$T$$
$13$ $$T + 4$$
$17$ $$T + 6$$
$19$ $$T - 1$$
$23$ $$T - 6$$
$29$ $$T + 6$$
$31$ $$T - 2$$
$37$ $$T + 4$$
$41$ $$T + 6$$
$43$ $$T + 4$$
$47$ $$T + 6$$
$53$ $$T + 6$$
$59$ $$T - 12$$
$61$ $$T - 14$$
$67$ $$T - 8$$
$71$ $$T$$
$73$ $$T - 14$$
$79$ $$T + 10$$
$83$ $$T - 12$$
$89$ $$T - 6$$
$97$ $$T + 10$$