Properties

 Label 350.2.a.a Level $350$ Weight $2$ Character orbit 350.a Self dual yes Analytic conductor $2.795$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$350 = 2 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 350.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$2.79476407074$$ Analytic rank: $$1$$ 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

 $$f(q)$$ $$=$$ $$q - q^{2} - q^{3} + q^{4} + q^{6} - q^{7} - q^{8} - 2q^{9} + O(q^{10})$$ $$q - q^{2} - q^{3} + q^{4} + q^{6} - q^{7} - q^{8} - 2q^{9} + 3q^{11} - q^{12} - 2q^{13} + q^{14} + q^{16} - 3q^{17} + 2q^{18} - 7q^{19} + q^{21} - 3q^{22} + q^{24} + 2q^{26} + 5q^{27} - q^{28} - 6q^{29} - 4q^{31} - q^{32} - 3q^{33} + 3q^{34} - 2q^{36} - 8q^{37} + 7q^{38} + 2q^{39} - 9q^{41} - q^{42} - 8q^{43} + 3q^{44} + 6q^{47} - q^{48} + q^{49} + 3q^{51} - 2q^{52} + 12q^{53} - 5q^{54} + q^{56} + 7q^{57} + 6q^{58} + 12q^{59} - 10q^{61} + 4q^{62} + 2q^{63} + q^{64} + 3q^{66} + 7q^{67} - 3q^{68} + 6q^{71} + 2q^{72} - 5q^{73} + 8q^{74} - 7q^{76} - 3q^{77} - 2q^{78} + 14q^{79} + q^{81} + 9q^{82} + 9q^{83} + q^{84} + 8q^{86} + 6q^{87} - 3q^{88} - 15q^{89} + 2q^{91} + 4q^{93} - 6q^{94} + q^{96} + 10q^{97} - q^{98} - 6q^{99} + O(q^{100})$$

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
−1.00000 −1.00000 1.00000 0 1.00000 −1.00000 −1.00000 −2.00000 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
$$2$$ $$1$$
$$5$$ $$1$$
$$7$$ $$1$$

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 350.2.a.a 1
3.b odd 2 1 3150.2.a.x 1
4.b odd 2 1 2800.2.a.x 1
5.b even 2 1 350.2.a.e yes 1
5.c odd 4 2 350.2.c.c 2
7.b odd 2 1 2450.2.a.m 1
15.d odd 2 1 3150.2.a.m 1
15.e even 4 2 3150.2.g.f 2
20.d odd 2 1 2800.2.a.h 1
20.e even 4 2 2800.2.g.i 2
35.c odd 2 1 2450.2.a.x 1
35.f even 4 2 2450.2.c.h 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
350.2.a.a 1 1.a even 1 1 trivial
350.2.a.e yes 1 5.b even 2 1
350.2.c.c 2 5.c odd 4 2
2450.2.a.m 1 7.b odd 2 1
2450.2.a.x 1 35.c odd 2 1
2450.2.c.h 2 35.f even 4 2
2800.2.a.h 1 20.d odd 2 1
2800.2.a.x 1 4.b odd 2 1
2800.2.g.i 2 20.e even 4 2
3150.2.a.m 1 15.d odd 2 1
3150.2.a.x 1 3.b odd 2 1
3150.2.g.f 2 15.e even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(350))$$:

 $$T_{3} + 1$$ $$T_{13} + 2$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T$$
$3$ $$1 + T$$
$5$ $$T$$
$7$ $$1 + T$$
$11$ $$-3 + T$$
$13$ $$2 + T$$
$17$ $$3 + T$$
$19$ $$7 + T$$
$23$ $$T$$
$29$ $$6 + T$$
$31$ $$4 + T$$
$37$ $$8 + T$$
$41$ $$9 + T$$
$43$ $$8 + T$$
$47$ $$-6 + T$$
$53$ $$-12 + T$$
$59$ $$-12 + T$$
$61$ $$10 + T$$
$67$ $$-7 + T$$
$71$ $$-6 + T$$
$73$ $$5 + T$$
$79$ $$-14 + T$$
$83$ $$-9 + T$$
$89$ $$15 + T$$
$97$ $$-10 + T$$